Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 4·5-s − 6-s − 7-s − 8-s − 2·9-s − 4·10-s − 11-s + 12-s + 13-s + 14-s + 4·15-s + 16-s + 4·17-s + 2·18-s + 2·19-s + 4·20-s − 21-s + 22-s − 7·23-s − 24-s + 11·25-s − 26-s − 5·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 1.26·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.970·17-s + 0.471·18-s + 0.458·19-s + 0.894·20-s − 0.218·21-s + 0.213·22-s − 1.45·23-s − 0.204·24-s + 11/5·25-s − 0.196·26-s − 0.962·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(182\)    =    \(2 \cdot 7 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{182} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 182,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.208975634$
$L(\frac12)$  $\approx$  $1.208975634$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.90738994183930, −18.48807794891326, −18.30342222642433, −17.01756600819702, −16.74809203016746, −15.40644834670140, −14.28972113089368, −13.75598355781158, −12.80366415530887, −11.49145261462399, −10.16850461541097, −9.706455892648276, −8.805211576902182, −7.730896020805833, −6.235123811216062, −5.545722002071029, −3.130731443960767, −1.899405535716451, 1.899405535716451, 3.130731443960767, 5.545722002071029, 6.235123811216062, 7.730896020805833, 8.805211576902182, 9.706455892648276, 10.16850461541097, 11.49145261462399, 12.80366415530887, 13.75598355781158, 14.28972113089368, 15.40644834670140, 16.74809203016746, 17.01756600819702, 18.30342222642433, 18.48807794891326, 19.90738994183930

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