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Properties

Label 2-3660-3660.1499-c0-0-0
Degree $2$
Conductor $3660$
Sign $-0.880 + 0.473i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.309 + 0.951i)3-s + (−0.104 − 0.994i)4-s + (−0.994 − 0.104i)5-s + (−0.499 − 0.866i)6-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.743 − 0.669i)10-s + (0.978 + 0.207i)12-s + (0.406 − 0.913i)15-s + (−0.978 + 0.207i)16-s + (−0.692 + 1.80i)17-s + (0.978 − 0.207i)18-s + (0.278 + 0.309i)19-s + i·20-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.309 + 0.951i)3-s + (−0.104 − 0.994i)4-s + (−0.994 − 0.104i)5-s + (−0.499 − 0.866i)6-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.743 − 0.669i)10-s + (0.978 + 0.207i)12-s + (0.406 − 0.913i)15-s + (−0.978 + 0.207i)16-s + (−0.692 + 1.80i)17-s + (0.978 − 0.207i)18-s + (0.278 + 0.309i)19-s + i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 61\)
Sign: $-0.880 + 0.473i$
Analytic conductor: \(1.82657\)
Root analytic conductor: \(1.35150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3660} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3660,\ (\ :0),\ -0.880 + 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3409351122\)
\(L(\frac12)\) \(\approx\) \(0.3409351122\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.669 - 0.743i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.994 + 0.104i)T \)
61 \( 1 + (0.406 + 0.913i)T \)
good7 \( 1 + (0.994 - 0.104i)T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.692 - 1.80i)T + (-0.743 - 0.669i)T^{2} \)
19 \( 1 + (-0.278 - 0.309i)T + (-0.104 + 0.994i)T^{2} \)
23 \( 1 + (-1.65 + 0.262i)T + (0.951 - 0.309i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T^{2} \)
31 \( 1 + (-0.685 - 1.05i)T + (-0.406 + 0.913i)T^{2} \)
37 \( 1 + (-0.587 - 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.743 - 0.669i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.84 + 0.292i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.406 - 0.913i)T^{2} \)
67 \( 1 + (-0.207 + 0.978i)T^{2} \)
71 \( 1 + (0.207 + 0.978i)T^{2} \)
73 \( 1 + (0.978 - 0.207i)T^{2} \)
79 \( 1 + (1.86 - 0.715i)T + (0.743 - 0.669i)T^{2} \)
83 \( 1 + (-0.413 - 1.94i)T + (-0.913 + 0.406i)T^{2} \)
89 \( 1 + (0.587 - 0.809i)T^{2} \)
97 \( 1 + (0.913 + 0.406i)T^{2} \)
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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

−8.987182642751449053505569908335, −8.383788397578421986479561391089, −7.951831777308973066696511015687, −6.75273126037143117693904434080, −6.42162960347068694172849721479, −5.29328990245356391732924262436, −4.74531633995892321812693304287, −3.96401774883402352644709289494, −3.03512931337064082545594591630, −1.34600873716073931441511683207, 0.29298030986127345739884771779, 1.37394029671026281640817388332, 2.76570477647167372070340684524, 3.06590545734824126885768566291, 4.47953758195477566558072649696, 5.01891149735695364660395637676, 6.42834913700042059532181949502, 7.12894884136875445192092234452, 7.53932689314145538570568423444, 8.231474357195028088817389313281

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