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Properties

Label 2-3620-3620.1819-c0-0-1
Degree $2$
Conductor $3620$
Sign $-0.593 - 0.804i$
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  + (0.374 − 0.927i)2-s + (1.18 − 0.628i)3-s + (−0.719 − 0.694i)4-s + (−0.978 − 0.207i)5-s + (−0.139 − 1.33i)6-s + (−0.939 − 1.62i)7-s + (−0.913 + 0.406i)8-s + (0.442 − 0.655i)9-s + (−0.559 + 0.829i)10-s + (−1.28 − 0.368i)12-s + (−1.86 + 0.261i)14-s + (−1.28 + 0.368i)15-s + (0.0348 + 0.999i)16-s + (−0.442 − 0.655i)18-s + (0.559 + 0.829i)20-s + (−2.13 − 1.33i)21-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s + (1.18 − 0.628i)3-s + (−0.719 − 0.694i)4-s + (−0.978 − 0.207i)5-s + (−0.139 − 1.33i)6-s + (−0.939 − 1.62i)7-s + (−0.913 + 0.406i)8-s + (0.442 − 0.655i)9-s + (−0.559 + 0.829i)10-s + (−1.28 − 0.368i)12-s + (−1.86 + 0.261i)14-s + (−1.28 + 0.368i)15-s + (0.0348 + 0.999i)16-s + (−0.442 − 0.655i)18-s + (0.559 + 0.829i)20-s + (−2.13 − 1.33i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3620\)    =    \(2^{2} \cdot 5 \cdot 181\)
Sign: $-0.593 - 0.804i$
Analytic conductor: \(1.80661\)
Root analytic conductor: \(1.34410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3620} (1819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3620,\ (\ :0),\ -0.593 - 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.093872064\)
\(L(\frac12)\) \(\approx\) \(1.093872064\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.374 + 0.927i)T \)
5 \( 1 + (0.978 + 0.207i)T \)
181 \( 1 + (0.719 + 0.694i)T \)
good3 \( 1 + (-1.18 + 0.628i)T + (0.559 - 0.829i)T^{2} \)
7 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.882 + 0.469i)T^{2} \)
13 \( 1 + (-0.961 - 0.275i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.76 - 0.123i)T + (0.990 - 0.139i)T^{2} \)
29 \( 1 + (-1.54 + 0.689i)T + (0.669 - 0.743i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.438 - 0.898i)T^{2} \)
41 \( 1 + (-0.671 + 1.37i)T + (-0.615 - 0.788i)T^{2} \)
43 \( 1 + (-0.213 - 1.21i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (1.75 + 0.503i)T + (0.848 + 0.529i)T^{2} \)
53 \( 1 + (0.615 - 0.788i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.671 + 0.563i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.204 + 1.94i)T + (-0.978 + 0.207i)T^{2} \)
71 \( 1 + (0.104 - 0.994i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.615 - 0.788i)T^{2} \)
83 \( 1 + (0.996 + 1.27i)T + (-0.241 + 0.970i)T^{2} \)
89 \( 1 + (1.10 + 0.924i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.882 - 0.469i)T^{2} \)
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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

−8.125467042942554435522809379495, −7.77398255897511672943233041125, −6.88011101735683021617303676298, −6.16338151596920342685284413740, −4.71565234677267624414058464431, −4.03703673018708438730963455585, −3.48631720325461186207815332859, −2.81538056198580764339402907780, −1.64084919936106449230104712741, −0.47019344937324784795721163511, 2.57778785034887601221577221278, 3.02375491071473447330409247080, 3.82220967687335338034377572181, 4.49247190263882498290026901751, 5.50058223500667334874661829614, 6.29441676970150023925259926301, 6.93678859929181195129288147670, 8.104696233369843179144883368131, 8.316250917745685067843919382580, 8.902785811382285422356079961306

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