Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 13^{2} $
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 + 0.554·3-s + 4-s + 1.35·5-s + 0.554·6-s − 7-s + 8-s − 2.69·9-s + 1.35·10-s + 1.80·11-s + 0.554·12-s − 14-s + 0.753·15-s + 16-s + 5.29·17-s − 2.69·18-s + 1.95·19-s + 1.35·20-s − 0.554·21-s + 1.80·22-s + 4.85·23-s + 0.554·24-s − 3.15·25-s − 3.15·27-s − 28-s + 5.96·29-s + 0.753·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.320·3-s + 0.5·4-s + 0.606·5-s + 0.226·6-s − 0.377·7-s + 0.353·8-s − 0.897·9-s + 0.429·10-s + 0.543·11-s + 0.160·12-s − 0.267·14-s + 0.194·15-s + 0.250·16-s + 1.28·17-s − 0.634·18-s + 0.447·19-s + 0.303·20-s − 0.121·21-s + 0.384·22-s + 1.01·23-s + 0.113·24-s − 0.631·25-s − 0.607·27-s − 0.188·28-s + 1.10·29-s + 0.137·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2366\)    =    \(2 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2366} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2366,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.510409636\)
\(L(\frac12)\)  \(\approx\)  \(3.510409636\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 - 0.554T + 3T^{2} \)
5 \( 1 - 1.35T + 5T^{2} \)
11 \( 1 - 1.80T + 11T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 1.95T + 19T^{2} \)
23 \( 1 - 4.85T + 23T^{2} \)
29 \( 1 - 5.96T + 29T^{2} \)
31 \( 1 + 3.63T + 31T^{2} \)
37 \( 1 - 7.75T + 37T^{2} \)
41 \( 1 - 0.0392T + 41T^{2} \)
43 \( 1 - 2.26T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 3.66T + 53T^{2} \)
59 \( 1 - 4.07T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 + 2.22T + 67T^{2} \)
71 \( 1 - 7.78T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 2.85T + 83T^{2} \)
89 \( 1 + 6.61T + 89T^{2} \)
97 \( 1 + 3.64T + 97T^{2} \)
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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

−9.139156178391628246486113408815, −8.122672452016226479949175296911, −7.39434499374166393153848589420, −6.43191263404335210442896837874, −5.79185349235949292139111204141, −5.18429092421811213887281254807, −4.01947128822591181734032450504, −3.16101185412149583870684179121, −2.48481634720681423580776757605, −1.14421363680449441349199471809, 1.14421363680449441349199471809, 2.48481634720681423580776757605, 3.16101185412149583870684179121, 4.01947128822591181734032450504, 5.18429092421811213887281254807, 5.79185349235949292139111204141, 6.43191263404335210442896837874, 7.39434499374166393153848589420, 8.122672452016226479949175296911, 9.139156178391628246486113408815

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