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Properties

Label 2-3660-3660.1019-c0-0-7
Degree $2$
Conductor $3660$
Sign $-0.869 + 0.494i$
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  + (0.207 − 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 + (0.281 − 0.434i)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 + (0.281 − 0.434i)17-s + (−0.866 − 0.5i)18-s + (1.81 + 0.809i)19-s + (−0.951 + 0.309i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 61\)
Sign: $-0.869 + 0.494i$
Analytic conductor: \(1.82657\)
Root analytic conductor: \(1.35150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3660} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3660,\ (\ :0),\ -0.869 + 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.966660774\)
\(L(\frac12)\) \(\approx\) \(1.966660774\)
\(L(1)\) 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 \( 1 + (-0.281 + 0.434i)T + (-0.406 - 0.913i)T^{2} \)
19 \( 1 + (-1.81 - 0.809i)T + (0.669 + 0.743i)T^{2} \)
23 \( 1 + (0.571 + 1.12i)T + (-0.587 + 0.809i)T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + (1.55 - 1.25i)T + (0.207 - 0.978i)T^{2} \)
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 \( 1 + (0.761 - 1.49i)T + (-0.587 - 0.809i)T^{2} \)
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 \( 1 + (1.56 - 1.01i)T + (0.406 - 0.913i)T^{2} \)
83 \( 1 + (1.47 - 0.155i)T + (0.978 - 0.207i)T^{2} \)
89 \( 1 + (0.951 + 0.309i)T^{2} \)
97 \( 1 + (-0.978 - 0.207i)T^{2} \)
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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

−8.696767850301025758214179237103, −7.86398769801601067314428130060, −7.04073483571072915810494104454, −5.91456426145701345162926443680, −5.37679682264174585480396178369, −4.41808521500477979087543172675, −3.48416656114974412198721239625, −2.74183384794379272240682909084, −1.78725882534030793235826395143, −1.05992191688337819443709644087, 1.77875459837716164629048006423, 3.05593073023833619235845624799, 3.52336236564359429528480592171, 4.52838821762417552368865133627, 5.48312642819548882911788414407, 5.83198990926692165290627640208, 7.03145832643369125721126253986, 7.48081946603807067108126586700, 8.127355176964835307889554556599, 9.141481679633612682358600660019

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