Properties

Degree 2
Conductor 131
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·3-s + 4-s − 1.61·5-s − 1.61·7-s − 0.618·9-s + 0.618·11-s + 0.618·12-s + 0.618·13-s − 1.00·15-s + 16-s − 1.61·20-s − 1.00·21-s + 1.61·25-s − 27-s − 1.61·28-s + 0.381·33-s + 2.61·35-s − 0.618·36-s + 0.381·39-s + 0.618·41-s − 1.61·43-s + 0.618·44-s + 0.999·45-s + 0.618·48-s + 1.61·49-s + 0.618·52-s + 2·53-s + ⋯
L(s)  = 1  + 0.618·3-s + 4-s − 1.61·5-s − 1.61·7-s − 0.618·9-s + 0.618·11-s + 0.618·12-s + 0.618·13-s − 1.00·15-s + 16-s − 1.61·20-s − 1.00·21-s + 1.61·25-s − 27-s − 1.61·28-s + 0.381·33-s + 2.61·35-s − 0.618·36-s + 0.381·39-s + 0.618·41-s − 1.61·43-s + 0.618·44-s + 0.999·45-s + 0.618·48-s + 1.61·49-s + 0.618·52-s + 2·53-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{131} (130, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 131,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.6403351049$
$L(\frac12)$  $\approx$  $0.6403351049$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 131$, \(F_p\) is a polynomial of degree 2. If $p = 131$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad131 \( 1 - T \)
good2 \( 1 - T^{2} \)
3 \( 1 - 0.618T + T^{2} \)
5 \( 1 + 1.61T + T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 - 0.618T + T^{2} \)
13 \( 1 - 0.618T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 2T + T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−13.49115710230681409798432755353, −12.29259752204418281575577204325, −11.66281031150107468242863614059, −10.63275133986296812825530566576, −9.218449254119487021935505079960, −8.157385197801971307650196111733, −7.11096588269597252147591005558, −6.14622486610113318867757722039, −3.75376533239769561770804369848, −3.03544317976724495517564615348, 3.03544317976724495517564615348, 3.75376533239769561770804369848, 6.14622486610113318867757722039, 7.11096588269597252147591005558, 8.157385197801971307650196111733, 9.218449254119487021935505079960, 10.63275133986296812825530566576, 11.66281031150107468242863614059, 12.29259752204418281575577204325, 13.49115710230681409798432755353

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