↑

Properties

Label 2-3660-3660.2519-c0-0-6
Degree $2$
Conductor $3660$
Sign $-0.232 + 0.972i$
Analytic cond. $1.82657$
Root an. cond. $1.35150$
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  + (−0.207 + 0.978i)2-s + (0.809 − 0.587i)3-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (0.406 + 0.913i)6-s + (0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (−0.5 − 0.866i)10-s + (−0.978 + 0.207i)12-s + (−0.207 + 0.978i)15-s + (0.669 + 0.743i)16-s + (−1.62 − 1.05i)17-s + (0.866 + 0.5i)18-s + (−1.81 − 0.809i)19-s + (0.951 − 0.309i)20-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (0.809 − 0.587i)3-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (0.406 + 0.913i)6-s + (0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (−0.5 − 0.866i)10-s + (−0.978 + 0.207i)12-s + (−0.207 + 0.978i)15-s + (0.669 + 0.743i)16-s + (−1.62 − 1.05i)17-s + (0.866 + 0.5i)18-s + (−1.81 − 0.809i)19-s + (0.951 − 0.309i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 61\)
Sign: $-0.232 + 0.972i$
Analytic conductor: \(1.82657\)
Root analytic conductor: \(1.35150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3660} (2519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3660,\ (\ :0),\ -0.232 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3579907034\)
\(L(\frac12)\) \(\approx\) \(0.3579907034\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.207 - 0.978i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.743 - 0.669i)T \)
61 \( 1 + (0.207 + 0.978i)T \)
good7 \( 1 + (-0.743 - 0.669i)T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (1.62 + 1.05i)T + (0.406 + 0.913i)T^{2} \)
19 \( 1 + (1.81 + 0.809i)T + (0.669 + 0.743i)T^{2} \)
23 \( 1 + (1.38 - 0.705i)T + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T^{2} \)
31 \( 1 + (0.0658 + 0.0813i)T + (-0.207 + 0.978i)T^{2} \)
37 \( 1 + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.406 + 0.913i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.970 - 0.494i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.207 - 0.978i)T^{2} \)
67 \( 1 + (-0.994 - 0.104i)T^{2} \)
71 \( 1 + (0.994 - 0.104i)T^{2} \)
73 \( 1 + (0.104 + 0.994i)T^{2} \)
79 \( 1 + (0.390 + 0.601i)T + (-0.406 + 0.913i)T^{2} \)
83 \( 1 + (1.47 - 0.155i)T + (0.978 - 0.207i)T^{2} \)
89 \( 1 + (-0.951 - 0.309i)T^{2} \)
97 \( 1 + (-0.978 - 0.207i)T^{2} \)
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

−8.314623013083077990095845285454, −7.79176218564371793522149025542, −6.88352821295355538294283593813, −6.74959956288886494813582735648, −5.84022496709796480676901090060, −4.41388510489770590365550644719, −4.14861767149062065385637218986, −2.92106184987984921369208415628, −1.99506254324202037648596189277, −0.18609943209770079647087747015, 1.78669205718282503263850352628, 2.38128730497811103902217303326, 3.69204218522518729738123196726, 4.16084855335761177115175492206, 4.55592546162120510173239827453, 5.71400377096260208735926254188, 6.89099780988684193156474737062, 8.044997109025652787385723747325, 8.472137509418173897792601509247, 8.705728062873955574165127599658

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