Properties

Degree 2
Conductor $ 5 \cdot 31 $
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 + 2·3-s − 4-s − 5-s − 2·6-s + 4·7-s + 3·8-s + 9-s + 10-s + 4·11-s − 2·12-s − 4·14-s − 2·15-s − 16-s − 8·17-s − 18-s + 4·19-s + 20-s + 8·21-s − 4·22-s + 2·23-s + 6·24-s + 25-s − 4·27-s − 4·28-s − 6·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.577·12-s − 1.06·14-s − 0.516·15-s − 1/4·16-s − 1.94·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 1.74·21-s − 0.852·22-s + 0.417·23-s + 1.22·24-s + 1/5·25-s − 0.769·27-s − 0.755·28-s − 1.11·29-s + 0.365·30-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  =  0
Selberg data  =  $(2,\ 155,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.057069688$
$L(\frac12)$  $\approx$  $1.057069688$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−19.31082846484836, −18.51519370386896, −17.56647688498951, −17.05241047449797, −15.56781650957480, −14.69329644985970, −14.05408199491893, −13.29030496383912, −11.67404196451840, −10.95042563470152, −9.406906855544248, −8.774417290846673, −8.125055688796013, −7.121812561758022, −4.892324402800400, −3.831082862999308, −1.767438441475689, 1.767438441475689, 3.831082862999308, 4.892324402800400, 7.121812561758022, 8.125055688796013, 8.774417290846673, 9.406906855544248, 10.95042563470152, 11.67404196451840, 13.29030496383912, 14.05408199491893, 14.69329644985970, 15.56781650957480, 17.05241047449797, 17.56647688498951, 18.51519370386896, 19.31082846484836

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