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 + 3·3-s + 4-s − 4·5-s + 3·6-s − 7-s + 8-s + 6·9-s − 4·10-s + 11-s + 3·12-s − 13-s − 14-s − 12·15-s + 16-s + 6·18-s − 6·19-s − 4·20-s − 3·21-s + 22-s − 7·23-s + 3·24-s + 11·25-s − 26-s + 9·27-s − 28-s − 4·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.78·5-s + 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 1.26·10-s + 0.301·11-s + 0.866·12-s − 0.277·13-s − 0.267·14-s − 3.09·15-s + 1/4·16-s + 1.41·18-s − 1.37·19-s − 0.894·20-s − 0.654·21-s + 0.213·22-s − 1.45·23-s + 0.612·24-s + 11/5·25-s − 0.196·26-s + 1.73·27-s − 0.188·28-s − 0.742·29-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  =  $(2,\ 182,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.135366401$
$L(\frac12)$  $\approx$  $2.135366401$
$L(\frac{3}{2})$   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 \( 1 - p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−19.88247752011130, −19.25377868977211, −18.62711522199385, −16.75088312096775, −15.74934755474166, −15.24756131521109, −14.61287591414215, −13.72057454122583, −12.68476149313497, −12.02186514366721, −10.77192769214820, −9.447300131639462, −8.238523811690696, −7.780552892986458, −6.627661893273811, −4.291243091362351, −3.806297220695687, −2.563810376497470, 2.563810376497470, 3.806297220695687, 4.291243091362351, 6.627661893273811, 7.780552892986458, 8.238523811690696, 9.447300131639462, 10.77192769214820, 12.02186514366721, 12.68476149313497, 13.72057454122583, 14.61287591414215, 15.24756131521109, 15.74934755474166, 16.75088312096775, 18.62711522199385, 19.25377868977211, 19.88247752011130

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