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Properties

Label 12-637e6-1.1-c1e6-0-0
Degree 1212
Conductor 6.681×10166.681\times 10^{16}
Sign 11
Analytic cond. 17318.017318.0
Root an. cond. 2.255322.25532
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·3-s + 2·4-s − 5·5-s − 8·6-s + 7·9-s − 10·10-s + 4·11-s − 8·12-s − 6·13-s + 20·15-s − 4·17-s + 14·18-s − 7·19-s − 10·20-s + 8·22-s − 23-s + 18·25-s − 12·26-s − 14·29-s + 40·30-s + 3·31-s + 4·32-s − 16·33-s − 8·34-s + 14·36-s + 10·37-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.30·3-s + 4-s − 2.23·5-s − 3.26·6-s + 7/3·9-s − 3.16·10-s + 1.20·11-s − 2.30·12-s − 1.66·13-s + 5.16·15-s − 0.970·17-s + 3.29·18-s − 1.60·19-s − 2.23·20-s + 1.70·22-s − 0.208·23-s + 18/5·25-s − 2.35·26-s − 2.59·29-s + 7.30·30-s + 0.538·31-s + 0.707·32-s − 2.78·33-s − 1.37·34-s + 7/3·36-s + 1.64·37-s + ⋯

Functional equation

Λ(s)=((712⋅136)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((712⋅136)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 712⋅1367^{12} \cdot 13^{6}
Sign: 11
Analytic conductor: 17318.017318.0
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 712⋅136, ( :[1/2]6), 1)(12,\ 7^{12} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.40388612470.4038861247
L(12)L(\frac12) ≈\approx 0.40388612470.4038861247
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)
bad7 1 1
13 (1+T)6 ( 1 + T )^{6}
good2 1−pT+pT2−p2T4+p2T5−p2T6+p3T7−p4T8+p5T10−p6T11+p6T12 1 - p T + p T^{2} - p^{2} T^{4} + p^{2} T^{5} - p^{2} T^{6} + p^{3} T^{7} - p^{4} T^{8} + p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12}
3 1+4T+p2T2+8T3−4pT4−62T5−137T6−62pT7−4p3T8+8p3T9+p6T10+4p5T11+p6T12 1 + 4 T + p^{2} T^{2} + 8 T^{3} - 4 p T^{4} - 62 T^{5} - 137 T^{6} - 62 p T^{7} - 4 p^{3} T^{8} + 8 p^{3} T^{9} + p^{6} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
5 1+pT+7T2+4T3+19T4−T5−146T6−pT7+19p2T8+4p3T9+7p4T10+p6T11+p6T12 1 + p T + 7 T^{2} + 4 T^{3} + 19 T^{4} - T^{5} - 146 T^{6} - p T^{7} + 19 p^{2} T^{8} + 4 p^{3} T^{9} + 7 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12}
11 1−4T−17T2+40T3+382T4−38pT5−3857T6−38p2T7+382p2T8+40p3T9−17p4T10−4p5T11+p6T12 1 - 4 T - 17 T^{2} + 40 T^{3} + 382 T^{4} - 38 p T^{5} - 3857 T^{6} - 38 p^{2} T^{7} + 382 p^{2} T^{8} + 40 p^{3} T^{9} - 17 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
17 1+4T−33T2−48T3+1080T4+470T5−20731T6+470pT7+1080p2T8−48p3T9−33p4T10+4p5T11+p6T12 1 + 4 T - 33 T^{2} - 48 T^{3} + 1080 T^{4} + 470 T^{5} - 20731 T^{6} + 470 p T^{7} + 1080 p^{2} T^{8} - 48 p^{3} T^{9} - 33 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
19 1+7T−5T2−28T3+469T4−875T5−18166T6−875pT7+469p2T8−28p3T9−5p4T10+7p5T11+p6T12 1 + 7 T - 5 T^{2} - 28 T^{3} + 469 T^{4} - 875 T^{5} - 18166 T^{6} - 875 p T^{7} + 469 p^{2} T^{8} - 28 p^{3} T^{9} - 5 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12}
23 1+T−27T2−150T3+51T4+1733T5+13694T6+1733pT7+51p2T8−150p3T9−27p4T10+p5T11+p6T12 1 + T - 27 T^{2} - 150 T^{3} + 51 T^{4} + 1733 T^{5} + 13694 T^{6} + 1733 p T^{7} + 51 p^{2} T^{8} - 150 p^{3} T^{9} - 27 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12}
29 (1+7T+74T2+403T3+74pT4+7p2T5+p3T6)2 ( 1 + 7 T + 74 T^{2} + 403 T^{3} + 74 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2}
31 1−3T−43T2+118T3+681T4−335T5−14122T6−335pT7+681p2T8+118p3T9−43p4T10−3p5T11+p6T12 1 - 3 T - 43 T^{2} + 118 T^{3} + 681 T^{4} - 335 T^{5} - 14122 T^{6} - 335 p T^{7} + 681 p^{2} T^{8} + 118 p^{3} T^{9} - 43 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
37 1−10T−19T2+126T3+3178T4−3932T5−126587T6−3932pT7+3178p2T8+126p3T9−19p4T10−10p5T11+p6T12 1 - 10 T - 19 T^{2} + 126 T^{3} + 3178 T^{4} - 3932 T^{5} - 126587 T^{6} - 3932 p T^{7} + 3178 p^{2} T^{8} + 126 p^{3} T^{9} - 19 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12}
41 (1−6T+7T2+12T3+7pT4−6p2T5+p3T6)2 ( 1 - 6 T + 7 T^{2} + 12 T^{3} + 7 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
43 (1−9T+124T2−673T3+124pT4−9p2T5+p3T6)2 ( 1 - 9 T + 124 T^{2} - 673 T^{3} + 124 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2}
47 1+17T+59T2+420T3+12425T4+71363T5+127970T6+71363pT7+12425p2T8+420p3T9+59p4T10+17p5T11+p6T12 1 + 17 T + 59 T^{2} + 420 T^{3} + 12425 T^{4} + 71363 T^{5} + 127970 T^{6} + 71363 p T^{7} + 12425 p^{2} T^{8} + 420 p^{3} T^{9} + 59 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12}
53 1+13T−29T2−164T3+10913T4+34735T5−380618T6+34735pT7+10913p2T8−164p3T9−29p4T10+13p5T11+p6T12 1 + 13 T - 29 T^{2} - 164 T^{3} + 10913 T^{4} + 34735 T^{5} - 380618 T^{6} + 34735 p T^{7} + 10913 p^{2} T^{8} - 164 p^{3} T^{9} - 29 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12}
59 1+22T+163T2+1366T3+21446T4+157474T5+757639T6+157474pT7+21446p2T8+1366p3T9+163p4T10+22p5T11+p6T12 1 + 22 T + 163 T^{2} + 1366 T^{3} + 21446 T^{4} + 157474 T^{5} + 757639 T^{6} + 157474 p T^{7} + 21446 p^{2} T^{8} + 1366 p^{3} T^{9} + 163 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12}
61 1−24T+233T2−1928T3+21078T4−166136T5+1069181T6−166136pT7+21078p2T8−1928p3T9+233p4T10−24p5T11+p6T12 1 - 24 T + 233 T^{2} - 1928 T^{3} + 21078 T^{4} - 166136 T^{5} + 1069181 T^{6} - 166136 p T^{7} + 21078 p^{2} T^{8} - 1928 p^{3} T^{9} + 233 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12}
67 1−14T+31T2+146T3−1022T4+44074T5−594409T6+44074pT7−1022p2T8+146p3T9+31p4T10−14p5T11+p6T12 1 - 14 T + 31 T^{2} + 146 T^{3} - 1022 T^{4} + 44074 T^{5} - 594409 T^{6} + 44074 p T^{7} - 1022 p^{2} T^{8} + 146 p^{3} T^{9} + 31 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12}
71 (1−4T+169T2−374T3+169pT4−4p2T5+p3T6)2 ( 1 - 4 T + 169 T^{2} - 374 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1+5T+25T2+1662T3+4155T4+5p2T5+1335370T6+5p3T7+4155p2T8+1662p3T9+25p4T10+5p5T11+p6T12 1 + 5 T + 25 T^{2} + 1662 T^{3} + 4155 T^{4} + 5 p^{2} T^{5} + 1335370 T^{6} + 5 p^{3} T^{7} + 4155 p^{2} T^{8} + 1662 p^{3} T^{9} + 25 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12}
79 1+T−167T2−346T3+14695T4+22873T5−1183810T6+22873pT7+14695p2T8−346p3T9−167p4T10+p5T11+p6T12 1 + T - 167 T^{2} - 346 T^{3} + 14695 T^{4} + 22873 T^{5} - 1183810 T^{6} + 22873 p T^{7} + 14695 p^{2} T^{8} - 346 p^{3} T^{9} - 167 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12}
83 (1−23T+376T2−4021T3+376pT4−23p2T5+p3T6)2 ( 1 - 23 T + 376 T^{2} - 4021 T^{3} + 376 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} )^{2}
89 1−11T−91T2+1626T3+4307T4−91583T5+367922T6−91583pT7+4307p2T8+1626p3T9−91p4T10−11p5T11+p6T12 1 - 11 T - 91 T^{2} + 1626 T^{3} + 4307 T^{4} - 91583 T^{5} + 367922 T^{6} - 91583 p T^{7} + 4307 p^{2} T^{8} + 1626 p^{3} T^{9} - 91 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12}
97 (1+3T+220T2+575T3+220pT4+3p2T5+p3T6)2 ( 1 + 3 T + 220 T^{2} + 575 T^{3} + 220 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
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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

−5.68603173571458807222172193263, −5.63950365973091143030525576815, −5.07571938015031445044820164257, −5.03660732250445029573639450357, −4.84600175098474077328786385672, −4.81116395692576187879968197717, −4.66594180245265279257420702879, −4.64691595290980830789121951508, −4.43055169186965200662577830710, −4.13365207863155134252920858907, −3.79277665833716766049605035926, −3.76513866867843848473978182481, −3.70627955530079397523053057805, −3.64314792404633435687460281059, −3.29069334475054533268532651372, −2.92005616738070379685673315941, −2.73043287485785794978781421092, −2.41278615077544224271228918786, −2.39634611083228809038662027370, −2.07830902742602164839761354412, −1.82461341601746379129452685230, −1.09603677917384233727651398137, −1.00785018681790447091423011225, −0.61706031465123877403041645720, −0.19524746124879706762616610652, 0.19524746124879706762616610652, 0.61706031465123877403041645720, 1.00785018681790447091423011225, 1.09603677917384233727651398137, 1.82461341601746379129452685230, 2.07830902742602164839761354412, 2.39634611083228809038662027370, 2.41278615077544224271228918786, 2.73043287485785794978781421092, 2.92005616738070379685673315941, 3.29069334475054533268532651372, 3.64314792404633435687460281059, 3.70627955530079397523053057805, 3.76513866867843848473978182481, 3.79277665833716766049605035926, 4.13365207863155134252920858907, 4.43055169186965200662577830710, 4.64691595290980830789121951508, 4.66594180245265279257420702879, 4.81116395692576187879968197717, 4.84600175098474077328786385672, 5.03660732250445029573639450357, 5.07571938015031445044820164257, 5.63950365973091143030525576815, 5.68603173571458807222172193263

Graph of the ZZ-function along the critical line

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