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Properties

Label 2-155-155.3-c1-0-10
Degree $2$
Conductor $155$
Sign $0.0176 + 0.999i$
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  + (0.106 − 0.671i)2-s + (−0.841 − 1.03i)3-s + (1.46 + 0.475i)4-s + (−2.21 + 0.313i)5-s + (−0.788 + 0.455i)6-s + (2.30 − 3.55i)7-s + (1.09 − 2.14i)8-s + (0.251 − 1.18i)9-s + (−0.0251 + 1.52i)10-s + (−1.45 + 1.31i)11-s + (−0.737 − 1.92i)12-s + (0.0161 − 0.0419i)13-s + (−2.13 − 1.92i)14-s + (2.18 + 2.03i)15-s + (1.16 + 0.845i)16-s + (−2.38 + 0.125i)17-s + ⋯
L(s)  = 1  + (0.0752 − 0.475i)2-s + (−0.486 − 0.600i)3-s + (0.731 + 0.237i)4-s + (−0.990 + 0.140i)5-s + (−0.321 + 0.185i)6-s + (0.871 − 1.34i)7-s + (0.386 − 0.758i)8-s + (0.0838 − 0.394i)9-s + (−0.00795 + 0.480i)10-s + (−0.439 + 0.395i)11-s + (−0.212 − 0.554i)12-s + (0.00447 − 0.0116i)13-s + (−0.571 − 0.514i)14-s + (0.565 + 0.526i)15-s + (0.290 + 0.211i)16-s + (−0.578 + 0.0303i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0176 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0176 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.0176 + 0.999i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ 0.0176 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783951 - 0.770207i\)
\(L(\frac12)\) \(\approx\) \(0.783951 - 0.770207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.21 - 0.313i)T \)
31 \( 1 + (-4.86 - 2.71i)T \)
good2 \( 1 + (-0.106 + 0.671i)T + (-1.90 - 0.618i)T^{2} \)
3 \( 1 + (0.841 + 1.03i)T + (-0.623 + 2.93i)T^{2} \)
7 \( 1 + (-2.30 + 3.55i)T + (-2.84 - 6.39i)T^{2} \)
11 \( 1 + (1.45 - 1.31i)T + (1.14 - 10.9i)T^{2} \)
13 \( 1 + (-0.0161 + 0.0419i)T + (-9.66 - 8.69i)T^{2} \)
17 \( 1 + (2.38 - 0.125i)T + (16.9 - 1.77i)T^{2} \)
19 \( 1 + (1.46 - 3.29i)T + (-12.7 - 14.1i)T^{2} \)
23 \( 1 + (-8.06 - 4.11i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (0.0681 - 0.0495i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + (0.208 - 0.778i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.526 - 5.01i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (5.85 - 2.24i)T + (31.9 - 28.7i)T^{2} \)
47 \( 1 + (8.99 - 1.42i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (0.133 - 0.0864i)T + (21.5 - 48.4i)T^{2} \)
59 \( 1 + (-10.2 - 1.07i)T + (57.7 + 12.2i)T^{2} \)
61 \( 1 - 0.733iT - 61T^{2} \)
67 \( 1 + (3.01 + 11.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.40 + 0.298i)T + (64.8 + 28.8i)T^{2} \)
73 \( 1 + (-8.42 - 0.441i)T + (72.6 + 7.63i)T^{2} \)
79 \( 1 + (11.2 - 12.5i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (-0.722 - 0.584i)T + (17.2 + 81.1i)T^{2} \)
89 \( 1 + (-5.38 + 16.5i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.65 + 3.24i)T + (-57.0 + 78.4i)T^{2} \)
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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

−12.55815203783958349323554150724, −11.49149115964508789792681440750, −11.13940624235539595968086427483, −10.11161848012326770762430098747, −8.147346517099331458051501474707, −7.29189375344158601410626341803, −6.68246310455800090835744931453, −4.63478834080525967461701626400, −3.41976321027278116241200865592, −1.29287857358478884741304078665, 2.52775114403578201999761697870, 4.74271058642048702624812266082, 5.35487106256222012510730027809, 6.77079154911206130827128458230, 8.040285506211619489241305495053, 8.786960593583497575394448808459, 10.57478660614760533207201833217, 11.28682849223420942459612881260, 11.77902972830516361230247124080, 13.12055883978939237937884382269

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