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 + 6-s + 7-s + 8-s − 2·9-s − 3·11-s + 12-s + 13-s + 14-s + 16-s − 2·18-s + 2·19-s + 21-s − 3·22-s − 3·23-s + 24-s − 5·25-s + 26-s − 5·27-s + 28-s + 5·31-s + 32-s − 3·33-s − 2·36-s − 7·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.471·18-s + 0.458·19-s + 0.218·21-s − 0.639·22-s − 0.625·23-s + 0.204·24-s − 25-s + 0.196·26-s − 0.962·27-s + 0.188·28-s + 0.898·31-s + 0.176·32-s − 0.522·33-s − 1/3·36-s − 1.15·37-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  =  $(2,\ 182,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.920406587$
$L(\frac12)$  $\approx$  $1.920406587$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−19.87687229986724, −19.12179982228050, −17.99244252026243, −17.17130645028664, −15.90284498908313, −15.39318758711295, −14.17738972652433, −13.86390444093683, −12.77123135624211, −11.73502655834995, −10.86871494678820, −9.687293312919998, −8.357542398475313, −7.633801709554254, −6.107431853137126, −5.065092345461235, −3.608584674187485, −2.330009733377074, 2.330009733377074, 3.608584674187485, 5.065092345461235, 6.107431853137126, 7.633801709554254, 8.357542398475313, 9.687293312919998, 10.86871494678820, 11.73502655834995, 12.77123135624211, 13.86390444093683, 14.17738972652433, 15.39318758711295, 15.90284498908313, 17.17130645028664, 17.99244252026243, 19.12179982228050, 19.87687229986724

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