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Properties

Label 12-3808e6-1.1-c1e6-0-2
Degree $12$
Conductor $3.049\times 10^{21}$
Sign $1$
Analytic cond. $7.90395\times 10^{8}$
Root an. cond. $5.51425$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 7^{6} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(7.90395\times 10^{8}\)
Root analytic conductor: \(5.51425\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.09235826\)
\(L(\frac12)\) \(\approx\) \(14.09235826\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line

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