Properties

Label 12-950e6-1.1-c1e6-0-3
Degree $12$
Conductor $7.351\times 10^{17}$
Sign $1$
Analytic cond. $190547.$
Root an. cond. $2.75423$
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  − 3·4-s − 4·9-s + 4·11-s + 6·16-s + 6·19-s − 28·29-s − 8·31-s + 12·36-s − 16·41-s − 12·44-s − 4·49-s + 12·59-s + 44·61-s − 10·64-s − 4·71-s − 18·76-s − 48·79-s − 12·81-s + 60·89-s − 16·99-s + 8·101-s + 4·109-s + 84·116-s + 10·121-s + 24·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 3/2·4-s − 4/3·9-s + 1.20·11-s + 3/2·16-s + 1.37·19-s − 5.19·29-s − 1.43·31-s + 2·36-s − 2.49·41-s − 1.80·44-s − 4/7·49-s + 1.56·59-s + 5.63·61-s − 5/4·64-s − 0.474·71-s − 2.06·76-s − 5.40·79-s − 4/3·81-s + 6.35·89-s − 1.60·99-s + 0.796·101-s + 0.383·109-s + 7.79·116-s + 0.909·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(190547.\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{950} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.419424682\)
\(L(\frac12)\) \(\approx\) \(1.419424682\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{2} )^{3} \)
5 \( 1 \)
19 \( ( 1 - T )^{6} \)
good3 \( 1 + 4 T^{2} + 28 T^{4} + 67 T^{6} + 28 p^{2} T^{8} + 4 p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 + 4 T^{2} + 12 p T^{4} + 3 p^{2} T^{6} + 12 p^{3} T^{8} + 4 p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 - 2 T + T^{2} - 20 T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 28 T^{2} + 380 T^{4} - 4369 T^{6} + 380 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 56 T^{2} + 1376 T^{4} - 24193 T^{6} + 1376 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 56 T^{2} + 1876 T^{4} - 47413 T^{6} + 1876 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 14 T + 128 T^{2} + 837 T^{3} + 128 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 4 T + 57 T^{2} + 96 T^{3} + 57 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 59 T^{2} + 686 T^{4} + 9553 T^{6} + 686 p^{2} T^{8} - 59 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 8 T + 75 T^{2} + 592 T^{3} + 75 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 174 T^{2} + 14599 T^{4} - 765892 T^{6} + 14599 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 63 T^{2} + 6238 T^{4} - 240059 T^{6} + 6238 p^{2} T^{8} - 63 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 240 T^{2} + 25780 T^{4} - 1682105 T^{6} + 25780 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 6 T + 148 T^{2} - 533 T^{3} + 148 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 22 T + 255 T^{2} - 2148 T^{3} + 255 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 140 T^{2} + 14280 T^{4} - 15495 p T^{6} + 14280 p^{2} T^{8} - 140 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 2 T + 197 T^{2} + 260 T^{3} + 197 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 240 T^{2} + 28680 T^{4} - 2348345 T^{6} + 28680 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 24 T + 349 T^{2} + 3472 T^{3} + 349 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 26 T^{2} + 10583 T^{4} - 158188 T^{6} + 10583 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 10 T + p T^{2} )^{6} \)
97 \( 1 - 222 T^{2} + 33855 T^{4} - 3795716 T^{6} + 33855 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.39537682621277548494733873925, −5.22960580558928601152261085071, −5.07510586676770463284290604887, −4.95351172314408700806764044554, −4.74746201180210820991398605848, −4.39348245550699026646732106531, −4.36182395210620741668243346284, −3.91575773019297067114119461641, −3.89379226034965580745316953999, −3.86774574820099920160136304989, −3.67831137126918413856069349132, −3.56189227257520263929000837317, −3.38198976849340266348304942073, −3.21113583497153760733090732349, −2.94648943131186639605656885630, −2.74498952232078345304881435295, −2.42777354679122028722842091150, −2.24223472969681158033100665910, −1.89896643138352464006313171727, −1.67398278921335836047318653017, −1.62392006404824758751436284376, −1.47669690046997311806703958247, −0.73686059420716282848172966438, −0.55640147237257796701151293681, −0.32428825887874405672795430583, 0.32428825887874405672795430583, 0.55640147237257796701151293681, 0.73686059420716282848172966438, 1.47669690046997311806703958247, 1.62392006404824758751436284376, 1.67398278921335836047318653017, 1.89896643138352464006313171727, 2.24223472969681158033100665910, 2.42777354679122028722842091150, 2.74498952232078345304881435295, 2.94648943131186639605656885630, 3.21113583497153760733090732349, 3.38198976849340266348304942073, 3.56189227257520263929000837317, 3.67831137126918413856069349132, 3.86774574820099920160136304989, 3.89379226034965580745316953999, 3.91575773019297067114119461641, 4.36182395210620741668243346284, 4.39348245550699026646732106531, 4.74746201180210820991398605848, 4.95351172314408700806764044554, 5.07510586676770463284290604887, 5.22960580558928601152261085071, 5.39537682621277548494733873925

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

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