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Properties

Label 2-3660-3660.1799-c0-0-3
Degree 22
Conductor 36603660
Sign 0.964−0.263i0.964 - 0.263i
Analytic cond. 1.826571.82657
Root an. cond. 1.351501.35150
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(3660s/2 ΓC(s) L(s)=((0.964−0.263i) Λ‾(1−s)\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}
Λ(s)=(3660s/2 ΓC(s) L(s)=((0.964−0.263i) Λ‾(1−s)\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: 22
Conductor: 36603660    =    22⋅3⋅5⋅612^{2} \cdot 3 \cdot 5 \cdot 61
Sign: 0.964−0.263i0.964 - 0.263i
Analytic conductor: 1.826571.82657
Root analytic conductor: 1.351501.35150
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3660(1799,⋅)\chi_{3660} (1799, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3660, ( :0), 0.964−0.263i)(2,\ 3660,\ (\ :0),\ 0.964 - 0.263i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.0960673392.096067339
L(12)L(\frac12) ≈\approx 2.0960673392.096067339
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.994−0.104i)T 1 + (-0.994 - 0.104i)T
3 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
5 1+(0.406−0.913i)T 1 + (0.406 - 0.913i)T
61 1+(−0.994+0.104i)T 1 + (-0.994 + 0.104i)T
good7 1+(−0.406−0.913i)T2 1 + (-0.406 - 0.913i)T^{2}
11 1+iT2 1 + iT^{2}
13 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1+(0.325−0.402i)T+(−0.207−0.978i)T2 1 + (0.325 - 0.402i)T + (-0.207 - 0.978i)T^{2}
19 1+(−1.45−0.309i)T+(0.913+0.406i)T2 1 + (-1.45 - 0.309i)T + (0.913 + 0.406i)T^{2}
23 1+(−0.103−0.0163i)T+(0.951+0.309i)T2 1 + (-0.103 - 0.0163i)T + (0.951 + 0.309i)T^{2}
29 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
31 1+(−0.715−0.0375i)T+(0.994+0.104i)T2 1 + (-0.715 - 0.0375i)T + (0.994 + 0.104i)T^{2}
37 1+(−0.587+0.809i)T2 1 + (-0.587 + 0.809i)T^{2}
41 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
43 1+(0.207−0.978i)T2 1 + (0.207 - 0.978i)T^{2}
47 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
53 1+(1.53−0.243i)T+(0.951−0.309i)T2 1 + (1.53 - 0.243i)T + (0.951 - 0.309i)T^{2}
59 1+(0.994−0.104i)T2 1 + (0.994 - 0.104i)T^{2}
67 1+(−0.743+0.669i)T2 1 + (-0.743 + 0.669i)T^{2}
71 1+(0.743+0.669i)T2 1 + (0.743 + 0.669i)T^{2}
73 1+(−0.669+0.743i)T2 1 + (-0.669 + 0.743i)T^{2}
79 1+(−0.846+0.685i)T+(0.207−0.978i)T2 1 + (-0.846 + 0.685i)T + (0.207 - 0.978i)T^{2}
83 1+(0.604+0.544i)T+(0.104+0.994i)T2 1 + (0.604 + 0.544i)T + (0.104 + 0.994i)T^{2}
89 1+(0.587+0.809i)T2 1 + (0.587 + 0.809i)T^{2}
97 1+(−0.104+0.994i)T2 1 + (-0.104 + 0.994i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line