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Properties

Label 2-3620-3620.3339-c0-0-1
Degree $2$
Conductor $3620$
Sign $0.365 - 0.930i$
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  + (0.719 + 0.694i)2-s + (0.116 − 0.173i)3-s + (0.0348 + 0.999i)4-s + (0.913 + 0.406i)5-s + (0.204 − 0.0434i)6-s + (0.766 − 1.32i)7-s + (−0.669 + 0.743i)8-s + (0.358 + 0.886i)9-s + (0.374 + 0.927i)10-s + (0.177 + 0.110i)12-s + (1.47 − 0.422i)14-s + (0.177 − 0.110i)15-s + (−0.997 + 0.0697i)16-s + (−0.358 + 0.886i)18-s + (−0.374 + 0.927i)20-s + (−0.140 − 0.287i)21-s + ⋯
L(s)  = 1  + (0.719 + 0.694i)2-s + (0.116 − 0.173i)3-s + (0.0348 + 0.999i)4-s + (0.913 + 0.406i)5-s + (0.204 − 0.0434i)6-s + (0.766 − 1.32i)7-s + (−0.669 + 0.743i)8-s + (0.358 + 0.886i)9-s + (0.374 + 0.927i)10-s + (0.177 + 0.110i)12-s + (1.47 − 0.422i)14-s + (0.177 − 0.110i)15-s + (−0.997 + 0.0697i)16-s + (−0.358 + 0.886i)18-s + (−0.374 + 0.927i)20-s + (−0.140 − 0.287i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3620\)    =    \(2^{2} \cdot 5 \cdot 181\)
Sign: $0.365 - 0.930i$
Analytic conductor: \(1.80661\)
Root analytic conductor: \(1.34410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3620} (3339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3620,\ (\ :0),\ 0.365 - 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.485668522\)
\(L(\frac12)\) \(\approx\) \(2.485668522\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.719 - 0.694i)T \)
5 \( 1 + (-0.913 - 0.406i)T \)
181 \( 1 + (-0.0348 - 0.999i)T \)
good3 \( 1 + (-0.116 + 0.173i)T + (-0.374 - 0.927i)T^{2} \)
7 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.559 - 0.829i)T^{2} \)
13 \( 1 + (-0.848 - 0.529i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.10 - 0.155i)T + (0.961 - 0.275i)T^{2} \)
29 \( 1 + (-0.586 + 0.651i)T + (-0.104 - 0.994i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.615 - 0.788i)T^{2} \)
41 \( 1 + (0.213 + 0.273i)T + (-0.241 + 0.970i)T^{2} \)
43 \( 1 + (0.454 - 0.165i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (1.13 + 0.709i)T + (0.438 + 0.898i)T^{2} \)
53 \( 1 + (0.241 + 0.970i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.213 - 1.21i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-1.78 + 0.379i)T + (0.913 - 0.406i)T^{2} \)
71 \( 1 + (0.978 + 0.207i)T^{2} \)
73 \( 1 + (0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.241 + 0.970i)T^{2} \)
83 \( 1 + (-0.149 + 0.599i)T + (-0.882 - 0.469i)T^{2} \)
89 \( 1 + (-0.0121 - 0.0687i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.559 + 0.829i)T^{2} \)
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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

−8.437184739555476026026027834054, −7.985881240622687130697491556981, −7.23555419489603630292106378312, −6.75176887397317349985377895099, −5.88082610939272627984100997914, −5.06340275471411982873976027607, −4.43656026685236667403201576284, −3.61284163010871877702551663732, −2.46477035302265156110046004085, −1.60998905762221953347227949818, 1.32809586603355764336835417078, 2.10685457428512555847357323078, 2.92451239580295712798056447774, 3.96194581330794228566555736887, 4.90433625810237071545236080867, 5.34475126363393059944570919299, 6.20572536795693856998552148192, 6.64742985856989337631236313330, 8.150469732913410052695758568526, 8.763495223307579493977746284880

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