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Properties

Label 12-3808e6-1.1-c1e6-0-2
Degree 1212
Conductor 3.049×10213.049\times 10^{21}
Sign 11
Analytic cond. 7.90395×1087.90395\times 10^{8}
Root an. cond. 5.514255.51425
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 6·7-s − 5·9-s − 4·11-s − 4·13-s − 6·17-s − 6·19-s − 3·25-s + 8·29-s + 12·31-s + 24·35-s + 10·37-s + 14·41-s − 12·43-s − 20·45-s + 2·47-s + 21·49-s + 26·53-s − 16·55-s + 22·59-s + 2·61-s − 30·63-s − 16·65-s − 14·67-s − 4·71-s + 18·73-s − 24·77-s + ⋯
L(s)  = 1  + 1.78·5-s + 2.26·7-s − 5/3·9-s − 1.20·11-s − 1.10·13-s − 1.45·17-s − 1.37·19-s − 3/5·25-s + 1.48·29-s + 2.15·31-s + 4.05·35-s + 1.64·37-s + 2.18·41-s − 1.82·43-s − 2.98·45-s + 0.291·47-s + 3·49-s + 3.57·53-s − 2.15·55-s + 2.86·59-s + 0.256·61-s − 3.77·63-s − 1.98·65-s − 1.71·67-s − 0.474·71-s + 2.10·73-s − 2.73·77-s + ⋯

Functional equation

Λ(s)=((230⋅76⋅176)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((230⋅76⋅176)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 230⋅76⋅1762^{30} \cdot 7^{6} \cdot 17^{6}
Sign: 11
Analytic conductor: 7.90395×1087.90395\times 10^{8}
Root analytic conductor: 5.514255.51425
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 230⋅76⋅176, ( :[1/2]6), 1)(12,\ 2^{30} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 14.0923582614.09235826
L(12)L(\frac12) ≈\approx 14.0923582614.09235826
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 (1−T)6 ( 1 - T )^{6}
17 (1+T)6 ( 1 + T )^{6}
good3 1+5T2+20T4+8T5+64T6+8pT7+20p2T8+5p4T10+p6T12 1 + 5 T^{2} + 20 T^{4} + 8 T^{5} + 64 T^{6} + 8 p T^{7} + 20 p^{2} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12}
5 1−4T+19T2−2p2T3+172T4−406T5+44p2T6−406pT7+172p2T8−2p5T9+19p4T10−4p5T11+p6T12 1 - 4 T + 19 T^{2} - 2 p^{2} T^{3} + 172 T^{4} - 406 T^{5} + 44 p^{2} T^{6} - 406 p T^{7} + 172 p^{2} T^{8} - 2 p^{5} T^{9} + 19 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
11 1+4T+40T2+156T3+763T4+2912T5+9752T6+2912pT7+763p2T8+156p3T9+40p4T10+4p5T11+p6T12 1 + 4 T + 40 T^{2} + 156 T^{3} + 763 T^{4} + 2912 T^{5} + 9752 T^{6} + 2912 p T^{7} + 763 p^{2} T^{8} + 156 p^{3} T^{9} + 40 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
13 1+4T+32T2+92T3+683T4+1720T5+10040T6+1720pT7+683p2T8+92p3T9+32p4T10+4p5T11+p6T12 1 + 4 T + 32 T^{2} + 92 T^{3} + 683 T^{4} + 1720 T^{5} + 10040 T^{6} + 1720 p T^{7} + 683 p^{2} T^{8} + 92 p^{3} T^{9} + 32 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
19 1+6T+106T2+514T3+4851T4+18492T5+121508T6+18492pT7+4851p2T8+514p3T9+106p4T10+6p5T11+p6T12 1 + 6 T + 106 T^{2} + 514 T^{3} + 4851 T^{4} + 18492 T^{5} + 121508 T^{6} + 18492 p T^{7} + 4851 p^{2} T^{8} + 514 p^{3} T^{9} + 106 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
23 1+86T2−96T3+3467T4−5712T5+93404T6−5712pT7+3467p2T8−96p3T9+86p4T10+p6T12 1 + 86 T^{2} - 96 T^{3} + 3467 T^{4} - 5712 T^{5} + 93404 T^{6} - 5712 p T^{7} + 3467 p^{2} T^{8} - 96 p^{3} T^{9} + 86 p^{4} T^{10} + p^{6} T^{12}
29 1−8T+168T2−1016T3+11691T4−54744T5+444296T6−54744pT7+11691p2T8−1016p3T9+168p4T10−8p5T11+p6T12 1 - 8 T + 168 T^{2} - 1016 T^{3} + 11691 T^{4} - 54744 T^{5} + 444296 T^{6} - 54744 p T^{7} + 11691 p^{2} T^{8} - 1016 p^{3} T^{9} + 168 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
31 1−12T+169T2−1340T3+11940T4−72636T5+478676T6−72636pT7+11940p2T8−1340p3T9+169p4T10−12p5T11+p6T12 1 - 12 T + 169 T^{2} - 1340 T^{3} + 11940 T^{4} - 72636 T^{5} + 478676 T^{6} - 72636 p T^{7} + 11940 p^{2} T^{8} - 1340 p^{3} T^{9} + 169 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}
37 1−10T+158T2−794T3+7235T4−15484T5+204204T6−15484pT7+7235p2T8−794p3T9+158p4T10−10p5T11+p6T12 1 - 10 T + 158 T^{2} - 794 T^{3} + 7235 T^{4} - 15484 T^{5} + 204204 T^{6} - 15484 p T^{7} + 7235 p^{2} T^{8} - 794 p^{3} T^{9} + 158 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12}
41 1−14T+239T2−2162T3+22152T4−153476T5+1169032T6−153476pT7+22152p2T8−2162p3T9+239p4T10−14p5T11+p6T12 1 - 14 T + 239 T^{2} - 2162 T^{3} + 22152 T^{4} - 153476 T^{5} + 1169032 T^{6} - 153476 p T^{7} + 22152 p^{2} T^{8} - 2162 p^{3} T^{9} + 239 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12}
43 1+12T+267T2+2214T3+27652T4+172218T5+1548504T6+172218pT7+27652p2T8+2214p3T9+267p4T10+12p5T11+p6T12 1 + 12 T + 267 T^{2} + 2214 T^{3} + 27652 T^{4} + 172218 T^{5} + 1548504 T^{6} + 172218 p T^{7} + 27652 p^{2} T^{8} + 2214 p^{3} T^{9} + 267 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}
47 1−2T+68T2−458T3+5539T4−23048T5+308656T6−23048pT7+5539p2T8−458p3T9+68p4T10−2p5T11+p6T12 1 - 2 T + 68 T^{2} - 458 T^{3} + 5539 T^{4} - 23048 T^{5} + 308656 T^{6} - 23048 p T^{7} + 5539 p^{2} T^{8} - 458 p^{3} T^{9} + 68 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
53 1−26T+561T2−7788T3+94040T4−868866T5+7109300T6−868866pT7+94040p2T8−7788p3T9+561p4T10−26p5T11+p6T12 1 - 26 T + 561 T^{2} - 7788 T^{3} + 94040 T^{4} - 868866 T^{5} + 7109300 T^{6} - 868866 p T^{7} + 94040 p^{2} T^{8} - 7788 p^{3} T^{9} + 561 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12}
59 1−22T+446T2−6234T3+73575T4−720764T5+5987812T6−720764pT7+73575p2T8−6234p3T9+446p4T10−22p5T11+p6T12 1 - 22 T + 446 T^{2} - 6234 T^{3} + 73575 T^{4} - 720764 T^{5} + 5987812 T^{6} - 720764 p T^{7} + 73575 p^{2} T^{8} - 6234 p^{3} T^{9} + 446 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12}
61 1−2T+235T2−1290T3+22528T4−199940T5+1453204T6−199940pT7+22528p2T8−1290p3T9+235p4T10−2p5T11+p6T12 1 - 2 T + 235 T^{2} - 1290 T^{3} + 22528 T^{4} - 199940 T^{5} + 1453204 T^{6} - 199940 p T^{7} + 22528 p^{2} T^{8} - 1290 p^{3} T^{9} + 235 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
67 1+14T+311T2+3190T3+43348T4+354524T5+3651136T6+354524pT7+43348p2T8+3190p3T9+311p4T10+14p5T11+p6T12 1 + 14 T + 311 T^{2} + 3190 T^{3} + 43348 T^{4} + 354524 T^{5} + 3651136 T^{6} + 354524 p T^{7} + 43348 p^{2} T^{8} + 3190 p^{3} T^{9} + 311 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12}
71 1+4T+222T2+724T3+26347T4+80736T5+2231500T6+80736pT7+26347p2T8+724p3T9+222p4T10+4p5T11+p6T12 1 + 4 T + 222 T^{2} + 724 T^{3} + 26347 T^{4} + 80736 T^{5} + 2231500 T^{6} + 80736 p T^{7} + 26347 p^{2} T^{8} + 724 p^{3} T^{9} + 222 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
73 1−18T+295T2−3662T3+37868T4−345492T5+3189936T6−345492pT7+37868p2T8−3662p3T9+295p4T10−18p5T11+p6T12 1 - 18 T + 295 T^{2} - 3662 T^{3} + 37868 T^{4} - 345492 T^{5} + 3189936 T^{6} - 345492 p T^{7} + 37868 p^{2} T^{8} - 3662 p^{3} T^{9} + 295 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12}
79 1+14T+304T2+4294T3+53683T4+544952T5+5699384T6+544952pT7+53683p2T8+4294p3T9+304p4T10+14p5T11+p6T12 1 + 14 T + 304 T^{2} + 4294 T^{3} + 53683 T^{4} + 544952 T^{5} + 5699384 T^{6} + 544952 p T^{7} + 53683 p^{2} T^{8} + 4294 p^{3} T^{9} + 304 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12}
83 1−8T+92T2−304T3+3195T4+75472T5−523856T6+75472pT7+3195p2T8−304p3T9+92p4T10−8p5T11+p6T12 1 - 8 T + 92 T^{2} - 304 T^{3} + 3195 T^{4} + 75472 T^{5} - 523856 T^{6} + 75472 p T^{7} + 3195 p^{2} T^{8} - 304 p^{3} T^{9} + 92 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
89 1−6T+152T2−1298T3+11307T4−28168T5+1079480T6−28168pT7+11307p2T8−1298p3T9+152p4T10−6p5T11+p6T12 1 - 6 T + 152 T^{2} - 1298 T^{3} + 11307 T^{4} - 28168 T^{5} + 1079480 T^{6} - 28168 p T^{7} + 11307 p^{2} T^{8} - 1298 p^{3} T^{9} + 152 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
97 1−20T+635T2−9326T3+161324T4−1773162T5+21172872T6−1773162pT7+161324p2T8−9326p3T9+635p4T10−20p5T11+p6T12 1 - 20 T + 635 T^{2} - 9326 T^{3} + 161324 T^{4} - 1773162 T^{5} + 21172872 T^{6} - 1773162 p T^{7} + 161324 p^{2} T^{8} - 9326 p^{3} T^{9} + 635 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.41354160169724615507728757838, −4.20810241207810829076160258075, −4.10140878561863739658955549193, −3.96489922628469560158487204407, −3.92253417032118084811303506311, −3.79900034025996332857232795601, −3.71713915686571034639885253720, −3.08926979906501893785924139370, −2.93788396227532454525846343632, −2.86565090084655429685845423603, −2.81514421503545396260763925825, −2.80515344890528305541842800254, −2.60139390952012692499057360149, −2.26334827753477923657006636715, −2.14986477984605906327721950078, −2.14578057839082416719690310075, −1.98845175238956406518744568915, −1.88969443124226177158445817433, −1.87257149522910020060418157941, −1.22492157377255172574602500417, −1.20661910779878850467317617513, −0.978791812451715055763874517882, −0.56183417445415962756432931572, −0.52074293022809453829544230120, −0.38502636708999702641485509839, 0.38502636708999702641485509839, 0.52074293022809453829544230120, 0.56183417445415962756432931572, 0.978791812451715055763874517882, 1.20661910779878850467317617513, 1.22492157377255172574602500417, 1.87257149522910020060418157941, 1.88969443124226177158445817433, 1.98845175238956406518744568915, 2.14578057839082416719690310075, 2.14986477984605906327721950078, 2.26334827753477923657006636715, 2.60139390952012692499057360149, 2.80515344890528305541842800254, 2.81514421503545396260763925825, 2.86565090084655429685845423603, 2.93788396227532454525846343632, 3.08926979906501893785924139370, 3.71713915686571034639885253720, 3.79900034025996332857232795601, 3.92253417032118084811303506311, 3.96489922628469560158487204407, 4.10140878561863739658955549193, 4.20810241207810829076160258075, 4.41354160169724615507728757838

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.