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 + 4-s + 2·5-s − 7-s + 8-s − 3·9-s + 2·10-s + 4·11-s − 13-s − 14-s + 16-s − 6·17-s − 3·18-s + 2·20-s + 4·22-s + 8·23-s − 25-s − 26-s − 28-s − 10·29-s − 8·31-s + 32-s − 6·34-s − 2·35-s − 3·36-s + 6·37-s + 2·40-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.632·10-s + 1.20·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.447·20-s + 0.852·22-s + 1.66·23-s − 1/5·25-s − 0.196·26-s − 0.188·28-s − 1.85·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.338·35-s − 1/2·36-s + 0.986·37-s + 0.316·40-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  =  $(2,\ 182,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.797169773$
$L(\frac12)$  $\approx$  $1.797169773$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.94243802880885, −19.09876349544161, −17.78665401228418, −17.06477932523864, −16.44541905890908, −14.96775717380866, −14.58470329421107, −13.42905230212440, −12.96795971682865, −11.58792978645654, −11.00197176121804, −9.508149438239248, −8.866721919717987, −7.088270553613740, −6.192176221889130, −5.215350578779274, −3.677643639441709, −2.187131809506494, 2.187131809506494, 3.677643639441709, 5.215350578779274, 6.192176221889130, 7.088270553613740, 8.866721919717987, 9.508149438239248, 11.00197176121804, 11.58792978645654, 12.96795971682865, 13.42905230212440, 14.58470329421107, 14.96775717380866, 16.44541905890908, 17.06477932523864, 17.78665401228418, 19.09876349544161, 19.94243802880885

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