Properties

Degree 2
Conductor $ 5 \cdot 31 $
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 − 5-s − 2·9-s − 4·11-s + 2·12-s − 6·13-s + 15-s + 4·16-s + 5·17-s − 19-s + 2·20-s + 8·23-s + 25-s + 5·27-s − 10·29-s − 31-s + 4·33-s + 4·36-s + 37-s + 6·39-s − 3·41-s − 7·43-s + 8·44-s + 2·45-s − 6·47-s − 4·48-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 2/3·9-s − 1.20·11-s + 0.577·12-s − 1.66·13-s + 0.258·15-s + 16-s + 1.21·17-s − 0.229·19-s + 0.447·20-s + 1.66·23-s + 1/5·25-s + 0.962·27-s − 1.85·29-s − 0.179·31-s + 0.696·33-s + 2/3·36-s + 0.164·37-s + 0.960·39-s − 0.468·41-s − 1.06·43-s + 1.20·44-s + 0.298·45-s − 0.875·47-s − 0.577·48-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(155\)    =    \(5 \cdot 31\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{155} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 155,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + T \)
31 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−19.50029733631710, −18.81585032133219, −18.00069668522337, −16.96269419992250, −16.65400473405299, −15.00245134623122, −14.59086631491092, −13.21710783727033, −12.53492345415150, −11.52172098246189, −10.40498491793852, −9.458542992010370, −8.232098142144936, −7.310464772394985, −5.448902117497666, −4.926918995975473, −3.143306602013360, 0, 3.143306602013360, 4.926918995975473, 5.448902117497666, 7.310464772394985, 8.232098142144936, 9.458542992010370, 10.40498491793852, 11.52172098246189, 12.53492345415150, 13.21710783727033, 14.59086631491092, 15.00245134623122, 16.65400473405299, 16.96269419992250, 18.00069668522337, 18.81585032133219, 19.50029733631710

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