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 − 0.801·3-s + 4-s − 3.04·5-s − 0.801·6-s − 7-s + 8-s − 2.35·9-s − 3.04·10-s − 1.24·11-s − 0.801·12-s − 14-s + 2.44·15-s + 16-s − 1.13·17-s − 2.35·18-s + 6.69·19-s − 3.04·20-s + 0.801·21-s − 1.24·22-s − 2.93·23-s − 0.801·24-s + 4.29·25-s + 4.29·27-s − 28-s − 4.54·29-s + 2.44·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.462·3-s + 0.5·4-s − 1.36·5-s − 0.327·6-s − 0.377·7-s + 0.353·8-s − 0.785·9-s − 0.964·10-s − 0.375·11-s − 0.231·12-s − 0.267·14-s + 0.631·15-s + 0.250·16-s − 0.275·17-s − 0.555·18-s + 1.53·19-s − 0.681·20-s + 0.174·21-s − 0.265·22-s − 0.612·23-s − 0.163·24-s + 0.859·25-s + 0.826·27-s − 0.188·28-s − 0.843·29-s + 0.446·30-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  =  \((2,\ 2366,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.307002919\)
\(L(\frac12)\)  \(\approx\)  \(1.307002919\)
\(L(\frac{3}{2})\)   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 \( 1 + 0.801T + 3T^{2} \)
5 \( 1 + 3.04T + 5T^{2} \)
11 \( 1 + 1.24T + 11T^{2} \)
17 \( 1 + 1.13T + 17T^{2} \)
19 \( 1 - 6.69T + 19T^{2} \)
23 \( 1 + 2.93T + 23T^{2} \)
29 \( 1 + 4.54T + 29T^{2} \)
31 \( 1 + 1.26T + 31T^{2} \)
37 \( 1 - 9.44T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 7.89T + 43T^{2} \)
47 \( 1 - 6.18T + 47T^{2} \)
53 \( 1 - 0.405T + 53T^{2} \)
59 \( 1 - 3.07T + 59T^{2} \)
61 \( 1 - 7.12T + 61T^{2} \)
67 \( 1 + 9.00T + 67T^{2} \)
71 \( 1 + 2.02T + 71T^{2} \)
73 \( 1 - 16.8T + 73T^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + 4.93T + 83T^{2} \)
89 \( 1 - 2.51T + 89T^{2} \)
97 \( 1 + 8.04T + 97T^{2} \)
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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

−8.910501463030750907306386368707, −7.83683571391805489191070037880, −7.57366572163028341905736455192, −6.53500235615994553079807682967, −5.73522778857034344744748916912, −5.04785349472335484044784897120, −4.07704227736615626234234253296, −3.40020875293993581659968882287, −2.50498974873250465675792636204, −0.65161611358665726622005258345, 0.65161611358665726622005258345, 2.50498974873250465675792636204, 3.40020875293993581659968882287, 4.07704227736615626234234253296, 5.04785349472335484044784897120, 5.73522778857034344744748916912, 6.53500235615994553079807682967, 7.57366572163028341905736455192, 7.83683571391805489191070037880, 8.910501463030750907306386368707

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