↑

Properties

Label 2-3380-3380.3119-c0-0-0
Degree $2$
Conductor $3380$
Sign $-0.613 - 0.790i$
Analytic cond. $1.68683$
Root an. cond. $1.29878$
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  + (−0.885 + 0.464i)2-s + (0.568 − 0.822i)4-s + (0.120 + 0.992i)5-s + (−0.120 + 0.992i)8-s + (−0.885 + 0.464i)9-s + (−0.568 − 0.822i)10-s + (0.120 + 0.992i)13-s + (−0.354 − 0.935i)16-s + (1.63 + 0.198i)17-s + (0.568 − 0.822i)18-s + (0.885 + 0.464i)20-s + (−0.970 + 0.239i)25-s + (−0.568 − 0.822i)26-s + (1.71 − 0.902i)29-s + (0.748 + 0.663i)32-s + ⋯
L(s)  = 1  + (−0.885 + 0.464i)2-s + (0.568 − 0.822i)4-s + (0.120 + 0.992i)5-s + (−0.120 + 0.992i)8-s + (−0.885 + 0.464i)9-s + (−0.568 − 0.822i)10-s + (0.120 + 0.992i)13-s + (−0.354 − 0.935i)16-s + (1.63 + 0.198i)17-s + (0.568 − 0.822i)18-s + (0.885 + 0.464i)20-s + (−0.970 + 0.239i)25-s + (−0.568 − 0.822i)26-s + (1.71 − 0.902i)29-s + (0.748 + 0.663i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-0.613 - 0.790i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (3119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ -0.613 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7301445642\)
\(L(\frac12)\) \(\approx\) \(0.7301445642\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.885 - 0.464i)T \)
5 \( 1 + (-0.120 - 0.992i)T \)
13 \( 1 + (-0.120 - 0.992i)T \)
good3 \( 1 + (0.885 - 0.464i)T^{2} \)
7 \( 1 + (0.970 + 0.239i)T^{2} \)
11 \( 1 + (0.568 + 0.822i)T^{2} \)
17 \( 1 + (-1.63 - 0.198i)T + (0.970 + 0.239i)T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-1.71 + 0.902i)T + (0.568 - 0.822i)T^{2} \)
31 \( 1 + (0.120 - 0.992i)T^{2} \)
37 \( 1 + (0.530 - 0.470i)T + (0.120 - 0.992i)T^{2} \)
41 \( 1 + (-0.222 - 0.902i)T + (-0.885 + 0.464i)T^{2} \)
43 \( 1 + (0.120 + 0.992i)T^{2} \)
47 \( 1 + (0.354 - 0.935i)T^{2} \)
53 \( 1 + (1.97 + 0.239i)T + (0.970 + 0.239i)T^{2} \)
59 \( 1 + (-0.748 - 0.663i)T^{2} \)
61 \( 1 + (-0.234 - 1.92i)T + (-0.970 + 0.239i)T^{2} \)
67 \( 1 + (0.354 - 0.935i)T^{2} \)
71 \( 1 + (0.885 - 0.464i)T^{2} \)
73 \( 1 + (1.77 + 0.929i)T + (0.568 + 0.822i)T^{2} \)
79 \( 1 + (0.354 - 0.935i)T^{2} \)
83 \( 1 + (-0.885 - 0.464i)T^{2} \)
89 \( 1 - 1.87iT - T^{2} \)
97 \( 1 + (-0.530 - 1.39i)T + (-0.748 + 0.663i)T^{2} \)
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

−9.030683431278842116147874906434, −8.071127358582288190957926504372, −7.82816884882445594462173379312, −6.77851156133101562225662800619, −6.27865417524294233327767382443, −5.59665830194541703920188073353, −4.63716970028917691264503013545, −3.25401182411022236105902592735, −2.54912615741238929960698502857, −1.44415148252525184296588706316, 0.61467770296203168875786660298, 1.58484902174047982122282055118, 2.98191512723398991450051504129, 3.41975457919605500056803012520, 4.71222765741094858105164633593, 5.57947473562830537586728842957, 6.25590767740215025788825409349, 7.35153730543126004803441680432, 8.107130582996127864660738426574, 8.516456223271239767464584039001

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