↑

Properties

Label 2-3620-3620.1979-c0-0-1
Degree $2$
Conductor $3620$
Sign $0.894 - 0.447i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.848 + 0.529i)2-s + (1.93 + 0.272i)3-s + (0.438 − 0.898i)4-s + (0.669 + 0.743i)5-s + (−1.78 + 0.795i)6-s + (−0.939 − 1.62i)7-s + (0.104 + 0.994i)8-s + (2.71 + 0.779i)9-s + (−0.961 − 0.275i)10-s + (1.09 − 1.62i)12-s + (1.65 + 0.882i)14-s + (1.09 + 1.62i)15-s + (−0.615 − 0.788i)16-s + (−2.71 + 0.779i)18-s + (0.961 − 0.275i)20-s + (−1.37 − 3.40i)21-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (1.93 + 0.272i)3-s + (0.438 − 0.898i)4-s + (0.669 + 0.743i)5-s + (−1.78 + 0.795i)6-s + (−0.939 − 1.62i)7-s + (0.104 + 0.994i)8-s + (2.71 + 0.779i)9-s + (−0.961 − 0.275i)10-s + (1.09 − 1.62i)12-s + (1.65 + 0.882i)14-s + (1.09 + 1.62i)15-s + (−0.615 − 0.788i)16-s + (−2.71 + 0.779i)18-s + (0.961 − 0.275i)20-s + (−1.37 − 3.40i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3620\)    =    \(2^{2} \cdot 5 \cdot 181\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(1.80661\)
Root analytic conductor: \(1.34410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3620} (1979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3620,\ (\ :0),\ 0.894 - 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.833043672\)
\(L(\frac12)\) \(\approx\) \(1.833043672\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.848 - 0.529i)T \)
5 \( 1 + (-0.669 - 0.743i)T \)
181 \( 1 + (-0.438 + 0.898i)T \)
good3 \( 1 + (-1.93 - 0.272i)T + (0.961 + 0.275i)T^{2} \)
7 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.990 + 0.139i)T^{2} \)
13 \( 1 + (-0.559 + 0.829i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.479 + 1.92i)T + (-0.882 - 0.469i)T^{2} \)
29 \( 1 + (-0.0783 - 0.745i)T + (-0.978 + 0.207i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.719 - 0.694i)T^{2} \)
41 \( 1 + (1.10 + 1.06i)T + (0.0348 + 0.999i)T^{2} \)
43 \( 1 + (0.0121 + 0.0687i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.116 + 0.173i)T + (-0.374 - 0.927i)T^{2} \)
53 \( 1 + (-0.0348 + 0.999i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.10 - 0.924i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (1.22 - 0.544i)T + (0.669 - 0.743i)T^{2} \)
71 \( 1 + (-0.913 - 0.406i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.0348 + 0.999i)T^{2} \)
83 \( 1 + (-0.0564 - 1.61i)T + (-0.997 + 0.0697i)T^{2} \)
89 \( 1 + (-0.671 - 0.563i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.990 - 0.139i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.851677210533353990416179774012, −8.097680548522689083413687545527, −7.18783947807133450517774244649, −7.03758030934309913766374135978, −6.31264149314380667191464673056, −4.84951192038299510174101025018, −3.92246744534582369769596425820, −3.09035679976531372548649324961, −2.40750373061684404100573773226, −1.28907982548226851223185463157, 1.52935532956760452560763400680, 2.10205533674125380739880364473, 2.99764314617236240517374175945, 3.38171487869549939909443466912, 4.62690435978723840990439419944, 5.89077025673795900668353035291, 6.67473986306573381967063247237, 7.61653560533601982343879023822, 8.243375900098201092081447657319, 8.864484687530335727605142278487

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