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Properties

Label 2-155-155.92-c1-0-4
Degree $2$
Conductor $155$
Sign $0.874 - 0.484i$
Analytic cond. $1.23768$
Root an. cond. $1.11251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.329 + 0.329i)2-s + (1.19 + 1.19i)3-s − 1.78i·4-s + (1.45 + 1.70i)5-s + 0.787i·6-s + (−1.32 − 1.32i)7-s + (1.24 − 1.24i)8-s − 0.152i·9-s + (−0.0818 + 1.04i)10-s + 6.44i·11-s + (2.12 − 2.12i)12-s + (−3.61 − 3.61i)13-s − 0.877i·14-s + (−0.296 + 3.76i)15-s − 2.74·16-s + (1.63 − 1.63i)17-s + ⋯
L(s)  = 1  + (0.233 + 0.233i)2-s + (0.688 + 0.688i)3-s − 0.891i·4-s + (0.649 + 0.760i)5-s + 0.321i·6-s + (−0.502 − 0.502i)7-s + (0.441 − 0.441i)8-s − 0.0508i·9-s + (−0.0258 + 0.328i)10-s + 1.94i·11-s + (0.613 − 0.613i)12-s + (−1.00 − 1.00i)13-s − 0.234i·14-s + (−0.0764 + 0.971i)15-s − 0.685·16-s + (0.396 − 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.874 - 0.484i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ 0.874 - 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50289 + 0.388694i\)
\(L(\frac12)\) \(\approx\) \(1.50289 + 0.388694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.45 - 1.70i)T \)
31 \( 1 + (-3.81 + 4.05i)T \)
good2 \( 1 + (-0.329 - 0.329i)T + 2iT^{2} \)
3 \( 1 + (-1.19 - 1.19i)T + 3iT^{2} \)
7 \( 1 + (1.32 + 1.32i)T + 7iT^{2} \)
11 \( 1 - 6.44iT - 11T^{2} \)
13 \( 1 + (3.61 + 3.61i)T + 13iT^{2} \)
17 \( 1 + (-1.63 + 1.63i)T - 17iT^{2} \)
19 \( 1 - 3.37iT - 19T^{2} \)
23 \( 1 + (2.82 + 2.82i)T + 23iT^{2} \)
29 \( 1 + 5.85T + 29T^{2} \)
37 \( 1 + (-0.296 + 0.296i)T - 37iT^{2} \)
41 \( 1 + 2.58T + 41T^{2} \)
43 \( 1 + (1.08 + 1.08i)T + 43iT^{2} \)
47 \( 1 + (2.65 + 2.65i)T + 47iT^{2} \)
53 \( 1 + (-3.91 - 3.91i)T + 53iT^{2} \)
59 \( 1 - 8.67iT - 59T^{2} \)
61 \( 1 - 7.52iT - 61T^{2} \)
67 \( 1 + (-8.64 - 8.64i)T + 67iT^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + (5.70 + 5.70i)T + 73iT^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 + (-12.0 - 12.0i)T + 83iT^{2} \)
89 \( 1 + 3.24T + 89T^{2} \)
97 \( 1 + (-0.522 - 0.522i)T + 97iT^{2} \)
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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

−13.31988372966733630515340044906, −12.18964139531890169813113913070, −10.36159079435161848131730557478, −9.984865025796026138793864825835, −9.528342358264476537890040639767, −7.54932580091238703588494417558, −6.63589134079606280665283765291, −5.31340977982749340799544425677, −4.01503165661050976173256769232, −2.38227639929550516603913466015, 2.14080768048088046484321686303, 3.34633519625353577429405244967, 5.09477262538388722532466197421, 6.49828062375032868095567582078, 7.892687915087358029121234693757, 8.647596406110022009554810824975, 9.459331452493225220882536470076, 11.17569151343490266917130333612, 12.17962116206422365902463150156, 12.97484236221158554784364959085

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