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Properties

Label 2-155-155.154-c0-0-0
Degree $2$
Conductor $155$
Sign $-0.5 - 0.866i$
Analytic cond. $0.0773550$
Root an. cond. $0.278127$
Motivic weight $0$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(0.0773550\)
Root analytic conductor: \(0.278127\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :0),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6078176341\)
\(L(\frac12)\) \(\approx\) \(0.6078176341\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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