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Properties

Label 2-3660-3660.1799-c0-0-3
Degree $2$
Conductor $3660$
Sign $0.964 - 0.263i$
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  + (0.994 + 0.104i)2-s + (−0.309 − 0.951i)3-s + (0.978 + 0.207i)4-s + (−0.406 + 0.913i)5-s + (−0.207 − 0.978i)6-s + (0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.499 + 0.866i)10-s + (−0.104 − 0.994i)12-s + (0.994 + 0.104i)15-s + (0.913 + 0.406i)16-s + (−0.325 + 0.402i)17-s + (−0.866 + 0.5i)18-s + (1.45 + 0.309i)19-s + (−0.587 + 0.809i)20-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (−0.309 − 0.951i)3-s + (0.978 + 0.207i)4-s + (−0.406 + 0.913i)5-s + (−0.207 − 0.978i)6-s + (0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s + (−0.499 + 0.866i)10-s + (−0.104 − 0.994i)12-s + (0.994 + 0.104i)15-s + (0.913 + 0.406i)16-s + (−0.325 + 0.402i)17-s + (−0.866 + 0.5i)18-s + (1.45 + 0.309i)19-s + (−0.587 + 0.809i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 61\)
Sign: $0.964 - 0.263i$
Analytic conductor: \(1.82657\)
Root analytic conductor: \(1.35150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3660} (1799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3660,\ (\ :0),\ 0.964 - 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.096067339\)
\(L(\frac12)\) \(\approx\) \(2.096067339\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.994 - 0.104i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.406 - 0.913i)T \)
61 \( 1 + (-0.994 + 0.104i)T \)
good7 \( 1 + (-0.406 - 0.913i)T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.325 - 0.402i)T + (-0.207 - 0.978i)T^{2} \)
19 \( 1 + (-1.45 - 0.309i)T + (0.913 + 0.406i)T^{2} \)
23 \( 1 + (-0.103 - 0.0163i)T + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T^{2} \)
31 \( 1 + (-0.715 - 0.0375i)T + (0.994 + 0.104i)T^{2} \)
37 \( 1 + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + (0.207 - 0.978i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.53 - 0.243i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.994 - 0.104i)T^{2} \)
67 \( 1 + (-0.743 + 0.669i)T^{2} \)
71 \( 1 + (0.743 + 0.669i)T^{2} \)
73 \( 1 + (-0.669 + 0.743i)T^{2} \)
79 \( 1 + (-0.846 + 0.685i)T + (0.207 - 0.978i)T^{2} \)
83 \( 1 + (0.604 + 0.544i)T + (0.104 + 0.994i)T^{2} \)
89 \( 1 + (0.587 + 0.809i)T^{2} \)
97 \( 1 + (-0.104 + 0.994i)T^{2} \)
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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

−8.282370542326186815039457672083, −7.69918597690318989608260852601, −7.16857415124556069776186729678, −6.46007030430190845862867762991, −5.90871245248483598027175370751, −5.09655774264093798218400383293, −4.10422261730467806534258260942, −3.14902738354187552439135137503, −2.54013604280408238543892076026, −1.39740595060115756532057462205, 1.01847965194732620556057739787, 2.54583849842729334281317633957, 3.50347287475603578058182959696, 4.12258520158076787303673606159, 5.05247522814020824106390056203, 5.21268357407293140594539852247, 6.18414326677547869525852164130, 7.07011763623169070991450731844, 7.908420426033585819926674198998, 8.756200797386247129193206846441

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