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Properties

Label 2-3620-3620.1799-c0-0-1
Degree $2$
Conductor $3620$
Sign $-0.925 - 0.379i$
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  + (0.882 − 0.469i)2-s + (−1.20 − 1.54i)3-s + (0.559 − 0.829i)4-s + (0.669 − 0.743i)5-s + (−1.78 − 0.795i)6-s + (0.766 − 1.32i)7-s + (0.104 − 0.994i)8-s + (−0.683 + 2.74i)9-s + (0.241 − 0.970i)10-s + (−1.95 + 0.136i)12-s + (0.0534 − 1.53i)14-s + (−1.95 − 0.136i)15-s + (−0.374 − 0.927i)16-s + (0.683 + 2.74i)18-s + (−0.241 − 0.970i)20-s + (−2.96 + 0.417i)21-s + ⋯
L(s)  = 1  + (0.882 − 0.469i)2-s + (−1.20 − 1.54i)3-s + (0.559 − 0.829i)4-s + (0.669 − 0.743i)5-s + (−1.78 − 0.795i)6-s + (0.766 − 1.32i)7-s + (0.104 − 0.994i)8-s + (−0.683 + 2.74i)9-s + (0.241 − 0.970i)10-s + (−1.95 + 0.136i)12-s + (0.0534 − 1.53i)14-s + (−1.95 − 0.136i)15-s + (−0.374 − 0.927i)16-s + (0.683 + 2.74i)18-s + (−0.241 − 0.970i)20-s + (−2.96 + 0.417i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3620\)    =    \(2^{2} \cdot 5 \cdot 181\)
Sign: $-0.925 - 0.379i$
Analytic conductor: \(1.80661\)
Root analytic conductor: \(1.34410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3620} (1799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3620,\ (\ :0),\ -0.925 - 0.379i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.696056995\)
\(L(\frac12)\) \(\approx\) \(1.696056995\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.882 + 0.469i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
181 \( 1 + (-0.559 + 0.829i)T \)
good3 \( 1 + (1.20 + 1.54i)T + (-0.241 + 0.970i)T^{2} \)
7 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.615 - 0.788i)T^{2} \)
13 \( 1 + (0.997 - 0.0697i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.885 - 0.855i)T + (0.0348 - 0.999i)T^{2} \)
29 \( 1 + (0.207 - 1.96i)T + (-0.978 - 0.207i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.961 + 0.275i)T^{2} \)
41 \( 1 + (-0.333 - 0.0957i)T + (0.848 + 0.529i)T^{2} \)
43 \( 1 + (-1.59 + 0.580i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.208 - 0.0145i)T + (0.990 - 0.139i)T^{2} \)
53 \( 1 + (-0.848 + 0.529i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.333 + 1.89i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (1.22 + 0.544i)T + (0.669 + 0.743i)T^{2} \)
71 \( 1 + (-0.913 + 0.406i)T^{2} \)
73 \( 1 + (0.939 + 0.342i)T^{2} \)
79 \( 1 + (-0.848 + 0.529i)T^{2} \)
83 \( 1 + (-1.37 - 0.857i)T + (0.438 + 0.898i)T^{2} \)
89 \( 1 + (-0.194 - 1.10i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (0.615 + 0.788i)T^{2} \)
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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

−7.82006068916917566267305395579, −7.46371771679975833461322373630, −6.60541247383874032365834597845, −6.04091130843871726078156789424, −5.15813034165262741674866125163, −4.89978121026247743255510509195, −3.78941226453099860298789185619, −2.22982045426320365653434115094, −1.51751380728509135123464962760, −0.889082364214780483242060201355, 2.21949785476693613852256925928, 3.01722284059161875627355198131, 4.15994514551699554023427150204, 4.61499126886574390744475472684, 5.57792273501744768042970312400, 5.91632557972943336820570458103, 6.29375911541704216324134622909, 7.45754314206524151645423223814, 8.522898322872759868549567706561, 9.190149790105368857917865422321

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