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Properties

Label 2-3620-3620.1099-c0-0-0
Degree $2$
Conductor $3620$
Sign $0.961 + 0.276i$
Analytic cond. $1.80661$
Root an. cond. $1.34410$
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  + (−0.961 + 0.275i)2-s + (−0.208 − 0.0145i)3-s + (0.848 − 0.529i)4-s + (0.913 + 0.406i)5-s + (0.204 − 0.0434i)6-s + (0.173 − 0.300i)7-s + (−0.669 + 0.743i)8-s + (−0.946 − 0.133i)9-s + (−0.990 − 0.139i)10-s + (−0.184 + 0.0981i)12-s + (−0.0840 + 0.336i)14-s + (−0.184 − 0.0981i)15-s + (0.438 − 0.898i)16-s + (0.946 − 0.133i)18-s + (0.990 − 0.139i)20-s + (−0.0406 + 0.0601i)21-s + ⋯
L(s)  = 1  + (−0.961 + 0.275i)2-s + (−0.208 − 0.0145i)3-s + (0.848 − 0.529i)4-s + (0.913 + 0.406i)5-s + (0.204 − 0.0434i)6-s + (0.173 − 0.300i)7-s + (−0.669 + 0.743i)8-s + (−0.946 − 0.133i)9-s + (−0.990 − 0.139i)10-s + (−0.184 + 0.0981i)12-s + (−0.0840 + 0.336i)14-s + (−0.184 − 0.0981i)15-s + (0.438 − 0.898i)16-s + (0.946 − 0.133i)18-s + (0.990 − 0.139i)20-s + (−0.0406 + 0.0601i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3620\)    =    \(2^{2} \cdot 5 \cdot 181\)
Sign: $0.961 + 0.276i$
Analytic conductor: \(1.80661\)
Root analytic conductor: \(1.34410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3620} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3620,\ (\ :0),\ 0.961 + 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8353473725\)
\(L(\frac12)\) \(\approx\) \(0.8353473725\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.961 - 0.275i)T \)
5 \( 1 + (-0.913 - 0.406i)T \)
181 \( 1 + (-0.848 + 0.529i)T \)
good3 \( 1 + (0.208 + 0.0145i)T + (0.990 + 0.139i)T^{2} \)
7 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.997 - 0.0697i)T^{2} \)
13 \( 1 + (0.882 - 0.469i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.22 + 1.57i)T + (-0.241 + 0.970i)T^{2} \)
29 \( 1 + (-0.748 + 0.831i)T + (-0.104 - 0.994i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.374 + 0.927i)T^{2} \)
41 \( 1 + (-0.704 + 1.74i)T + (-0.719 - 0.694i)T^{2} \)
43 \( 1 + (-1.10 - 0.924i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-1.18 + 0.628i)T + (0.559 - 0.829i)T^{2} \)
53 \( 1 + (0.719 - 0.694i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.704 + 0.256i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (-1.78 + 0.379i)T + (0.913 - 0.406i)T^{2} \)
71 \( 1 + (0.978 + 0.207i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.719 - 0.694i)T^{2} \)
83 \( 1 + (-0.444 - 0.429i)T + (0.0348 + 0.999i)T^{2} \)
89 \( 1 + (1.59 + 0.580i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (0.997 + 0.0697i)T^{2} \)
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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

−8.658241713996945533805639340805, −8.097449752854578406317166903439, −7.20849858123697990851863822021, −6.43510801333360974772396552062, −5.95710558867615640840696048357, −5.29099952311973744134095054234, −4.07095684831634754320803370893, −2.66431911619004379322585578442, −2.23397255985006750910934797464, −0.77698246645841822538652771116, 1.12229650853664067971727237086, 2.16136747163591496764244138231, 2.86837911574337628621127897279, 4.01243200469538530555171483036, 5.30453947149542934607290708228, 5.79585674683305867057754695744, 6.51408853643394165734142931792, 7.46585755260763988923844246691, 8.274331174472405548852892961162, 8.758580688343841288211586619916

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