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Properties

Label 2-155-155.64-c1-0-6
Degree $2$
Conductor $155$
Sign $0.888 - 0.457i$
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  + (−1.59 + 2.19i)2-s + (0.0633 + 0.0871i)3-s + (−1.65 − 5.08i)4-s + (0.708 − 2.12i)5-s − 0.292·6-s + (1.26 − 0.412i)7-s + (8.62 + 2.80i)8-s + (0.923 − 2.84i)9-s + (3.52 + 4.93i)10-s + (0.645 + 1.98i)11-s + (0.338 − 0.466i)12-s + (−3.10 − 4.27i)13-s + (−1.11 + 3.43i)14-s + (0.229 − 0.0725i)15-s + (−11.2 + 8.15i)16-s + (0.510 + 0.166i)17-s + ⋯
L(s)  = 1  + (−1.12 + 1.55i)2-s + (0.0365 + 0.0503i)3-s + (−0.825 − 2.54i)4-s + (0.316 − 0.948i)5-s − 0.119·6-s + (0.479 − 0.155i)7-s + (3.04 + 0.990i)8-s + (0.307 − 0.947i)9-s + (1.11 + 1.55i)10-s + (0.194 + 0.599i)11-s + (0.0977 − 0.134i)12-s + (−0.861 − 1.18i)13-s + (−0.298 + 0.919i)14-s + (0.0593 − 0.0187i)15-s + (−2.80 + 2.03i)16-s + (0.123 + 0.0402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.888 - 0.457i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ 0.888 - 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661116 + 0.160283i\)
\(L(\frac12)\) \(\approx\) \(0.661116 + 0.160283i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.708 + 2.12i)T \)
31 \( 1 + (1.02 + 5.47i)T \)
good2 \( 1 + (1.59 - 2.19i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.0633 - 0.0871i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.26 + 0.412i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.645 - 1.98i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.10 + 4.27i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.510 - 0.166i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.33 - 0.968i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-4.84 - 1.57i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.49 - 3.99i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 9.47iT - 37T^{2} \)
41 \( 1 + (4.83 + 3.51i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.57 + 7.66i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (0.330 + 0.454i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.593 + 0.192i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.23 - 5.98i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 7.07T + 61T^{2} \)
67 \( 1 - 9.26iT - 67T^{2} \)
71 \( 1 + (1.57 - 4.84i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.22 - 1.37i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.81 + 8.66i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.19 - 3.01i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.98 - 9.19i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.26 + 0.736i)T + (78.4 - 57.0i)T^{2} \)
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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

−13.27847495857053391815818203369, −12.11332388352954281255949024378, −10.34274277105464434893922157296, −9.647127562173854357672508225634, −8.798215463316015904169060492715, −7.83286068556198908167372060308, −6.89273161845656613697308801631, −5.59406668357706362233065051155, −4.71557221515599506986711492858, −1.08405420088987874151930829451, 1.88711196112669103093530315547, 2.98738804122019343190798273014, 4.63260102355256834822950652736, 6.97329950918231565856473825382, 7.958722648553188629066887094744, 9.105101224274718883833319214807, 9.986281103519451741148887972941, 10.94375289290993202374417986724, 11.41911496230395153547846210447, 12.50290176126272632001055690725

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