Properties

Degree 2
Conductor $ 5 \cdot 31 $
Sign $-0.5 + 0.866i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·2-s − 1.99·4-s + (0.5 − 0.866i)5-s + 1.73i·7-s + 1.73i·8-s − 9-s + (−1.49 − 0.866i)10-s + 2.99·14-s + 0.999·16-s + 1.73i·18-s + 19-s + (−0.999 + 1.73i)20-s + (−0.499 − 0.866i)25-s − 3.46i·28-s − 31-s + ⋯
L(s)  = 1  − 1.73i·2-s − 1.99·4-s + (0.5 − 0.866i)5-s + 1.73i·7-s + 1.73i·8-s − 9-s + (−1.49 − 0.866i)10-s + 2.99·14-s + 0.999·16-s + 1.73i·18-s + 19-s + (−0.999 + 1.73i)20-s + (−0.499 − 0.866i)25-s − 3.46i·28-s − 31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(155\)    =    \(5 \cdot 31\)
\( \varepsilon \)  =  $-0.5 + 0.866i$
motivic weight  =  \(0\)
character  :  $\chi_{155} (154, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 155,\ (\ :0),\ -0.5 + 0.866i)$
$L(\frac{1}{2})$  $\approx$  $0.6078176341$
$L(\frac12)$  $\approx$  $0.6078176341$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;31\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{5,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
good2 \( 1 + 1.73iT - T^{2} \)
3 \( 1 + T^{2} \)
7 \( 1 - 1.73iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−12.46543236559967768776943439337, −11.95316490223993661237857601247, −11.15479282252388540651046991477, −9.749128852540363752407603347420, −9.069309539962566150968174134441, −8.406564653898943386903774874381, −5.79099571878796810866265396472, −4.99559637066076860419486785128, −3.14553519709855962013572144934, −1.98470118185236666196526716097, 3.57720643177974959481399999539, 5.14454408149836053986851032894, 6.28352037519677019263430211854, 7.16240109251553910414852154377, 7.87127156377874814137322110235, 9.234470675715962152120679315755, 10.30377698260116748695414024712, 11.32697481286746022943537966114, 13.24129811164260988978612131150, 14.01728269487878139040519980875

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