Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 2·5-s − 7-s − 2·9-s + 2·12-s − 3·13-s + 2·15-s + 4·16-s + 4·17-s − 2·19-s + 4·20-s + 21-s − 2·23-s − 25-s + 5·27-s + 2·28-s − 2·31-s + 2·35-s + 4·36-s − 8·37-s + 3·39-s − 3·41-s + 3·43-s + 4·45-s + 10·47-s − 4·48-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.894·5-s − 0.377·7-s − 2/3·9-s + 0.577·12-s − 0.832·13-s + 0.516·15-s + 16-s + 0.970·17-s − 0.458·19-s + 0.894·20-s + 0.218·21-s − 0.417·23-s − 1/5·25-s + 0.962·27-s + 0.377·28-s − 0.359·31-s + 0.338·35-s + 2/3·36-s − 1.31·37-s + 0.480·39-s − 0.468·41-s + 0.457·43-s + 0.596·45-s + 1.45·47-s − 0.577·48-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(131\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{131} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 131,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 131$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 131$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad131 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−19.93448937840128, −19.26656784092465, −18.48419675491006, −17.35807708686373, −16.85812044749217, −15.74534125980926, −14.64862679593371, −13.86396617834908, −12.48457426181911, −12.03123178923170, −10.74854190018750, −9.665115817523667, −8.516334285594620, −7.508141702839985, −5.901932766074342, −4.773053986093166, −3.441795603679315, 0, 3.441795603679315, 4.773053986093166, 5.901932766074342, 7.508141702839985, 8.516334285594620, 9.665115817523667, 10.74854190018750, 12.03123178923170, 12.48457426181911, 13.86396617834908, 14.64862679593371, 15.74534125980926, 16.85812044749217, 17.35807708686373, 18.48419675491006, 19.26656784092465, 19.93448937840128

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