Properties

Degree $12$
Conductor $7.089\times 10^{18}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·4-s − 2·8-s + 3·11-s − 9·16-s + 3·17-s + 3·19-s + 9·22-s + 9·23-s + 6·25-s − 18·29-s − 6·31-s − 9·32-s + 9·34-s − 9·37-s + 9·38-s − 24·41-s + 18·43-s + 9·44-s + 27·46-s − 15·47-s − 12·49-s + 18·50-s − 54·58-s + 9·59-s + 6·61-s − 18·62-s + ⋯
L(s)  = 1  + 2.12·2-s + 3/2·4-s − 0.707·8-s + 0.904·11-s − 9/4·16-s + 0.727·17-s + 0.688·19-s + 1.91·22-s + 1.87·23-s + 6/5·25-s − 3.34·29-s − 1.07·31-s − 1.59·32-s + 1.54·34-s − 1.47·37-s + 1.45·38-s − 3.74·41-s + 2.74·43-s + 1.35·44-s + 3.98·46-s − 2.18·47-s − 1.71·49-s + 2.54·50-s − 7.09·58-s + 1.17·59-s + 0.768·61-s − 2.28·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.06044318380\)
\(L(\frac12)\) \(\approx\) \(0.06044318380\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} )^{3} \)
3 \( 1 \)
7 \( 1 + 12 T^{2} - 4 T^{3} + 12 p T^{4} + p^{3} T^{6} \)
11 \( ( 1 - T + T^{2} )^{3} \)
good5 \( 1 - 6 T^{2} - 8 T^{3} + 6 T^{4} + 24 T^{5} + 86 T^{6} + 24 p T^{7} + 6 p^{2} T^{8} - 8 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 + 3 T^{2} + 32 T^{3} + 3 p T^{4} + p^{3} T^{6} )^{2} \)
17 \( ( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
19 \( 1 - 3 T - 12 T^{2} + 237 T^{3} - 408 T^{4} - 1839 T^{5} + 24590 T^{6} - 1839 p T^{7} - 408 p^{2} T^{8} + 237 p^{3} T^{9} - 12 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 9 T + 9 T^{2} + 6 p T^{3} - 9 T^{4} - 3249 T^{5} + 16702 T^{6} - 3249 p T^{7} - 9 p^{2} T^{8} + 6 p^{4} T^{9} + 9 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 + 9 T + 3 p T^{2} + 430 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 6 T + 39 T^{2} + 262 T^{3} + 942 T^{4} + 7302 T^{5} + 44307 T^{6} + 7302 p T^{7} + 942 p^{2} T^{8} + 262 p^{3} T^{9} + 39 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 9 T - 18 T^{2} - 113 T^{3} + 1620 T^{4} - 3915 T^{5} - 120216 T^{6} - 3915 p T^{7} + 1620 p^{2} T^{8} - 113 p^{3} T^{9} - 18 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 12 T + 144 T^{2} + 22 p T^{3} + 144 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 9 T + 117 T^{2} - 610 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 15 T + 33 T^{2} + 98 T^{3} + 6975 T^{4} + 29487 T^{5} - 87106 T^{6} + 29487 p T^{7} + 6975 p^{2} T^{8} + 98 p^{3} T^{9} + 33 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 123 T^{2} - 64 T^{3} + 8610 T^{4} + 3936 T^{5} - 503059 T^{6} + 3936 p T^{7} + 8610 p^{2} T^{8} - 64 p^{3} T^{9} - 123 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 - 9 T - 141 T^{3} - 864 T^{4} + 30879 T^{5} - 171866 T^{6} + 30879 p T^{7} - 864 p^{2} T^{8} - 141 p^{3} T^{9} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 18 T^{2} + 688 T^{3} - 3414 T^{4} - 10122 T^{5} + 398166 T^{6} - 10122 p T^{7} - 3414 p^{2} T^{8} + 688 p^{3} T^{9} - 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T - 150 T^{2} - 324 T^{3} + 17814 T^{4} + 11310 T^{5} - 1359610 T^{6} + 11310 p T^{7} + 17814 p^{2} T^{8} - 324 p^{3} T^{9} - 150 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + 3 T + 189 T^{2} + 362 T^{3} + 189 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 123 T^{2} - 384 T^{3} + 6150 T^{4} + 23616 T^{5} - 289519 T^{6} + 23616 p T^{7} + 6150 p^{2} T^{8} - 384 p^{3} T^{9} - 123 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 + 12 T - 60 T^{2} - 608 T^{3} + 7164 T^{4} - 180 T^{5} - 865506 T^{6} - 180 p T^{7} + 7164 p^{2} T^{8} - 608 p^{3} T^{9} - 60 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 6 T + 138 T^{2} - 1184 T^{3} + 138 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 36 T + 633 T^{2} + 8396 T^{3} + 102066 T^{4} + 1165620 T^{5} + 11868377 T^{6} + 1165620 p T^{7} + 102066 p^{2} T^{8} + 8396 p^{3} T^{9} + 633 p^{4} T^{10} + 36 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 9 T + 210 T^{2} - 1753 T^{3} + 210 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.13420629089399528464973569721, −4.94061394688219252873275905449, −4.55928918583613051558644840741, −4.53453936305293463289359683384, −4.46059525284850912912553513194, −4.43680300918022817724780497086, −4.04199420513766855944301247050, −3.88576389447668866589578455121, −3.72369387986976542221537858866, −3.58672772644776853918932395368, −3.32835945909119646539991780145, −3.27304800428326881138336107871, −3.21015468608521083762286225699, −3.20083201620226236588913291561, −2.90470292820128859807903846959, −2.61022382274300642236050991269, −2.46337452148674481529389724608, −1.88456246931673577459305186051, −1.85066185270786591950555710697, −1.79920726663430106721304094073, −1.70527046813312249582710211818, −1.27280395697751210010120275782, −0.811291647395531261001553647214, −0.78269220352060580761730316496, −0.02089584621481332332997949204, 0.02089584621481332332997949204, 0.78269220352060580761730316496, 0.811291647395531261001553647214, 1.27280395697751210010120275782, 1.70527046813312249582710211818, 1.79920726663430106721304094073, 1.85066185270786591950555710697, 1.88456246931673577459305186051, 2.46337452148674481529389724608, 2.61022382274300642236050991269, 2.90470292820128859807903846959, 3.20083201620226236588913291561, 3.21015468608521083762286225699, 3.27304800428326881138336107871, 3.32835945909119646539991780145, 3.58672772644776853918932395368, 3.72369387986976542221537858866, 3.88576389447668866589578455121, 4.04199420513766855944301247050, 4.43680300918022817724780497086, 4.46059525284850912912553513194, 4.53453936305293463289359683384, 4.55928918583613051558644840741, 4.94061394688219252873275905449, 5.13420629089399528464973569721

Graph of the $Z$-function along the critical line

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