Properties

Degree 2
Conductor 11197
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 4-s − 2·5-s + 3·6-s − 5·7-s + 3·8-s + 6·9-s + 2·10-s − 4·11-s + 3·12-s − 7·13-s + 5·14-s + 6·15-s − 16-s − 2·17-s − 6·18-s − 5·19-s + 2·20-s + 15·21-s + 4·22-s − 2·23-s − 9·24-s − 25-s + 7·26-s − 9·27-s + 5·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 1/2·4-s − 0.894·5-s + 1.22·6-s − 1.88·7-s + 1.06·8-s + 2·9-s + 0.632·10-s − 1.20·11-s + 0.866·12-s − 1.94·13-s + 1.33·14-s + 1.54·15-s − 1/4·16-s − 0.485·17-s − 1.41·18-s − 1.14·19-s + 0.447·20-s + 3.27·21-s + 0.852·22-s − 0.417·23-s − 1.83·24-s − 1/5·25-s + 1.37·26-s − 1.73·27-s + 0.944·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11197\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{11197} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 11197,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 11197$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 11197$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11197 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−17.30355194604508, −16.69544248412029, −16.47880123802363, −15.78736162609968, −15.48467302042307, −14.71872970001793, −13.59164529057994, −13.03294065622619, −12.62621399213507, −12.22307129572691, −11.60870356674085, −10.73251051031271, −10.27521719178064, −10.00867682320660, −9.384352916495701, −8.528251728745352, −7.736943781142023, −7.064728465829807, −6.799274899613764, −5.878394085356669, −5.174340169346392, −4.655601092025473, −3.961578927392342, −3.029214703782400, −1.855407758031725, 0, 0, 0, 1.855407758031725, 3.029214703782400, 3.961578927392342, 4.655601092025473, 5.174340169346392, 5.878394085356669, 6.799274899613764, 7.064728465829807, 7.736943781142023, 8.528251728745352, 9.384352916495701, 10.00867682320660, 10.27521719178064, 10.73251051031271, 11.60870356674085, 12.22307129572691, 12.62621399213507, 13.03294065622619, 13.59164529057994, 14.71872970001793, 15.48467302042307, 15.78736162609968, 16.47880123802363, 16.69544248412029, 17.30355194604508

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