Properties

Label 12-637e6-1.1-c1e6-0-0
Degree $12$
Conductor $6.681\times 10^{16}$
Sign $1$
Analytic cond. $17318.0$
Root an. cond. $2.25532$
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  + 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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(7^{12} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(17318.0\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 7^{12} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4038861247\)
\(L(\frac12)\) \(\approx\) \(0.4038861247\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( ( 1 + T )^{6} \)
good2 \( 1 - p T + p T^{2} - p^{2} T^{4} + p^{2} T^{5} - p^{2} T^{6} + p^{3} T^{7} - p^{4} T^{8} + p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \)
3 \( 1 + 4 T + p^{2} T^{2} + 8 T^{3} - 4 p T^{4} - 62 T^{5} - 137 T^{6} - 62 p T^{7} - 4 p^{3} T^{8} + 8 p^{3} T^{9} + p^{6} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + p T + 7 T^{2} + 4 T^{3} + 19 T^{4} - T^{5} - 146 T^{6} - p T^{7} + 19 p^{2} T^{8} + 4 p^{3} T^{9} + 7 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 - 4 T - 17 T^{2} + 40 T^{3} + 382 T^{4} - 38 p T^{5} - 3857 T^{6} - 38 p^{2} T^{7} + 382 p^{2} T^{8} + 40 p^{3} T^{9} - 17 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T - 33 T^{2} - 48 T^{3} + 1080 T^{4} + 470 T^{5} - 20731 T^{6} + 470 p T^{7} + 1080 p^{2} T^{8} - 48 p^{3} T^{9} - 33 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 7 T - 5 T^{2} - 28 T^{3} + 469 T^{4} - 875 T^{5} - 18166 T^{6} - 875 p T^{7} + 469 p^{2} T^{8} - 28 p^{3} T^{9} - 5 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + T - 27 T^{2} - 150 T^{3} + 51 T^{4} + 1733 T^{5} + 13694 T^{6} + 1733 p T^{7} + 51 p^{2} T^{8} - 150 p^{3} T^{9} - 27 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 7 T + 74 T^{2} + 403 T^{3} + 74 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 - 3 T - 43 T^{2} + 118 T^{3} + 681 T^{4} - 335 T^{5} - 14122 T^{6} - 335 p T^{7} + 681 p^{2} T^{8} + 118 p^{3} T^{9} - 43 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 10 T - 19 T^{2} + 126 T^{3} + 3178 T^{4} - 3932 T^{5} - 126587 T^{6} - 3932 p T^{7} + 3178 p^{2} T^{8} + 126 p^{3} T^{9} - 19 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 - 6 T + 7 T^{2} + 12 T^{3} + 7 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 9 T + 124 T^{2} - 673 T^{3} + 124 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 17 T + 59 T^{2} + 420 T^{3} + 12425 T^{4} + 71363 T^{5} + 127970 T^{6} + 71363 p T^{7} + 12425 p^{2} T^{8} + 420 p^{3} T^{9} + 59 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 13 T - 29 T^{2} - 164 T^{3} + 10913 T^{4} + 34735 T^{5} - 380618 T^{6} + 34735 p T^{7} + 10913 p^{2} T^{8} - 164 p^{3} T^{9} - 29 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 22 T + 163 T^{2} + 1366 T^{3} + 21446 T^{4} + 157474 T^{5} + 757639 T^{6} + 157474 p T^{7} + 21446 p^{2} T^{8} + 1366 p^{3} T^{9} + 163 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 24 T + 233 T^{2} - 1928 T^{3} + 21078 T^{4} - 166136 T^{5} + 1069181 T^{6} - 166136 p T^{7} + 21078 p^{2} T^{8} - 1928 p^{3} T^{9} + 233 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 14 T + 31 T^{2} + 146 T^{3} - 1022 T^{4} + 44074 T^{5} - 594409 T^{6} + 44074 p T^{7} - 1022 p^{2} T^{8} + 146 p^{3} T^{9} + 31 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 4 T + 169 T^{2} - 374 T^{3} + 169 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 5 T + 25 T^{2} + 1662 T^{3} + 4155 T^{4} + 5 p^{2} T^{5} + 1335370 T^{6} + 5 p^{3} T^{7} + 4155 p^{2} T^{8} + 1662 p^{3} T^{9} + 25 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + T - 167 T^{2} - 346 T^{3} + 14695 T^{4} + 22873 T^{5} - 1183810 T^{6} + 22873 p T^{7} + 14695 p^{2} T^{8} - 346 p^{3} T^{9} - 167 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 23 T + 376 T^{2} - 4021 T^{3} + 376 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 11 T - 91 T^{2} + 1626 T^{3} + 4307 T^{4} - 91583 T^{5} + 367922 T^{6} - 91583 p T^{7} + 4307 p^{2} T^{8} + 1626 p^{3} T^{9} - 91 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 3 T + 220 T^{2} + 575 T^{3} + 220 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line

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