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Properties

Label 2-3660-3660.1499-c0-0-6
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

\(L(\frac{1}{2})\) \(\approx\) \(2.225267479\)
\(L(\frac12)\) \(\approx\) \(2.225267479\)
\(L(1)\) 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

−8.520835418675228056055675298903, −7.48576775251704706060254663148, −6.82841026589873577202575760372, −5.97362261223308726267977152087, −5.53704208226566623344243482955, −4.62780473204381418907939399088, −3.37603191978107673667251945673, −2.73514877329721288835417982230, −1.93930216876560395610471382687, −1.00939866272458158434239695742, 1.99732315808733553506832139194, 2.87921827755711657415228620231, 3.91773278623148982903527795648, 4.36297608999466799395601712396, 5.50881907575141353155388127412, 5.78538132333117834917616725708, 6.51795405663168902208065540479, 7.64163484833410241200954920387, 8.370292709270745375902396691453, 8.788994492071571794328288351098

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