Properties

Label 2-855-5.4-c1-0-30
Degree $2$
Conductor $855$
Sign $0.165 + 0.986i$
Analytic cond. $6.82720$
Root an. cond. $2.61289$
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  − 0.906i·2-s + 1.17·4-s + (0.370 + 2.20i)5-s − 2.59i·7-s − 2.88i·8-s + (1.99 − 0.336i)10-s − 0.741·11-s − 3.78i·13-s − 2.35·14-s − 0.258·16-s − 3.16i·17-s + 19-s + (0.436 + 2.59i)20-s + 0.672i·22-s − 0.570i·23-s + ⋯
L(s)  = 1  − 0.641i·2-s + 0.588·4-s + (0.165 + 0.986i)5-s − 0.981i·7-s − 1.01i·8-s + (0.632 − 0.106i)10-s − 0.223·11-s − 1.05i·13-s − 0.629·14-s − 0.0647·16-s − 0.768i·17-s + 0.229·19-s + (0.0975 + 0.580i)20-s + 0.143i·22-s − 0.119i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(855\)    =    \(3^{2} \cdot 5 \cdot 19\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(6.82720\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{855} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 855,\ (\ :1/2),\ 0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43716 - 1.21581i\)
\(L(\frac12)\) \(\approx\) \(1.43716 - 1.21581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.370 - 2.20i)T \)
19 \( 1 - T \)
good2 \( 1 + 0.906iT - 2T^{2} \)
7 \( 1 + 2.59iT - 7T^{2} \)
11 \( 1 + 0.741T + 11T^{2} \)
13 \( 1 + 3.78iT - 13T^{2} \)
17 \( 1 + 3.16iT - 17T^{2} \)
23 \( 1 + 0.570iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 5.83T + 31T^{2} \)
37 \( 1 - 1.40iT - 37T^{2} \)
41 \( 1 - 3.83T + 41T^{2} \)
43 \( 1 + 2.59iT - 43T^{2} \)
47 \( 1 - 5.08iT - 47T^{2} \)
53 \( 1 - 0.160iT - 53T^{2} \)
59 \( 1 - 8.35T + 59T^{2} \)
61 \( 1 + 8.57T + 61T^{2} \)
67 \( 1 - 14.8iT - 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + 1.83T + 79T^{2} \)
83 \( 1 - 4.19iT - 83T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 - 3.78iT - 97T^{2} \)
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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

−10.32468240158380320972306879611, −9.603501358918820012999657483820, −8.094663516375591439809703847252, −7.30763250343128350611339042101, −6.70846533133356913665145977568, −5.72675792090165696003312939635, −4.31798559317634937561398688010, −3.17644940240857250388479928074, −2.58514641659851077324678392742, −0.961246311443921929651030259647, 1.63373839324785596726698187873, 2.67431927381163926755411113721, 4.32289491246926012575726101773, 5.29521076006401585573314191413, 6.02149632613426477450951275997, 6.78761352205866707793526320130, 7.969544884928370259141898104442, 8.540503832087814769025826280019, 9.281037133292102684033768338719, 10.27103432802632473901376876810

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