Properties

Label 2-2-1.1-c7-0-0
Degree $2$
Conductor $2$
Sign $1$
Analytic cond. $0.624770$
Root an. cond. $0.790423$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 12·3-s + 64·4-s − 210·5-s − 96·6-s + 1.01e3·7-s − 512·8-s − 2.04e3·9-s + 1.68e3·10-s + 1.09e3·11-s + 768·12-s + 1.38e3·13-s − 8.12e3·14-s − 2.52e3·15-s + 4.09e3·16-s + 1.47e4·17-s + 1.63e4·18-s − 3.99e4·19-s − 1.34e4·20-s + 1.21e4·21-s − 8.73e3·22-s + 6.87e4·23-s − 6.14e3·24-s − 3.40e4·25-s − 1.10e4·26-s − 5.07e4·27-s + 6.50e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.256·3-s + 1/2·4-s − 0.751·5-s − 0.181·6-s + 1.11·7-s − 0.353·8-s − 0.934·9-s + 0.531·10-s + 0.247·11-s + 0.128·12-s + 0.174·13-s − 0.791·14-s − 0.192·15-s + 1/4·16-s + 0.725·17-s + 0.660·18-s − 1.33·19-s − 0.375·20-s + 0.287·21-s − 0.174·22-s + 1.17·23-s − 0.0907·24-s − 0.435·25-s − 0.123·26-s − 0.496·27-s + 0.559·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2\)
Sign: $1$
Analytic conductor: \(0.624770\)
Root analytic conductor: \(0.790423\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.6812925309\)
\(L(\frac12)\) \(\approx\) \(0.6812925309\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{3} T \)
good3 \( 1 - 4 p T + p^{7} T^{2} \)
5 \( 1 + 42 p T + p^{7} T^{2} \)
7 \( 1 - 1016 T + p^{7} T^{2} \)
11 \( 1 - 1092 T + p^{7} T^{2} \)
13 \( 1 - 1382 T + p^{7} T^{2} \)
17 \( 1 - 14706 T + p^{7} T^{2} \)
19 \( 1 + 39940 T + p^{7} T^{2} \)
23 \( 1 - 68712 T + p^{7} T^{2} \)
29 \( 1 + 102570 T + p^{7} T^{2} \)
31 \( 1 - 227552 T + p^{7} T^{2} \)
37 \( 1 - 160526 T + p^{7} T^{2} \)
41 \( 1 - 10842 T + p^{7} T^{2} \)
43 \( 1 + 630748 T + p^{7} T^{2} \)
47 \( 1 - 472656 T + p^{7} T^{2} \)
53 \( 1 + 1494018 T + p^{7} T^{2} \)
59 \( 1 - 2640660 T + p^{7} T^{2} \)
61 \( 1 - 827702 T + p^{7} T^{2} \)
67 \( 1 + 126004 T + p^{7} T^{2} \)
71 \( 1 + 1414728 T + p^{7} T^{2} \)
73 \( 1 - 980282 T + p^{7} T^{2} \)
79 \( 1 + 3566800 T + p^{7} T^{2} \)
83 \( 1 - 5672892 T + p^{7} T^{2} \)
89 \( 1 + 11951190 T + p^{7} T^{2} \)
97 \( 1 - 8682146 T + p^{7} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.15692226908151683971050874281, −27.00357744914318573616419510009, −25.27268835228433640069786954447, −23.46593741983178606917145905174, −20.82742945542969457904045414740, −19.21245663157084290808201817048, −17.17832430504048038861703275990, −14.86169320152756118045171000251, −11.39598699301533616475028388759, −8.272040919984955894587034898992, 8.272040919984955894587034898992, 11.39598699301533616475028388759, 14.86169320152756118045171000251, 17.17832430504048038861703275990, 19.21245663157084290808201817048, 20.82742945542969457904045414740, 23.46593741983178606917145905174, 25.27268835228433640069786954447, 27.00357744914318573616419510009, 28.15692226908151683971050874281

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