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Properties

Label 2-1332-148.139-c0-0-1
Degree 22
Conductor 13321332
Sign 0.815+0.578i0.815 + 0.578i
Analytic cond. 0.6647540.664754
Root an. cond. 0.8153240.815324
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  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (0.223 − 0.266i)5-s + (0.866 − 0.5i)8-s + (0.173 − 0.300i)10-s + (−0.592 − 1.62i)13-s + (0.766 − 0.642i)16-s + (−0.524 + 1.43i)17-s + (0.118 − 0.326i)20-s + (0.152 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (−1.62 + 0.939i)29-s + (0.642 − 0.766i)32-s + (−0.266 + 1.50i)34-s + (0.939 − 0.342i)37-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (0.223 − 0.266i)5-s + (0.866 − 0.5i)8-s + (0.173 − 0.300i)10-s + (−0.592 − 1.62i)13-s + (0.766 − 0.642i)16-s + (−0.524 + 1.43i)17-s + (0.118 − 0.326i)20-s + (0.152 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (−1.62 + 0.939i)29-s + (0.642 − 0.766i)32-s + (−0.266 + 1.50i)34-s + (0.939 − 0.342i)37-s + ⋯

Functional equation

Λ(s)=(1332s/2 ΓC(s) L(s)=((0.815+0.578i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1332s/2 ΓC(s) L(s)=((0.815+0.578i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13321332    =    22⋅32⋅372^{2} \cdot 3^{2} \cdot 37
Sign: 0.815+0.578i0.815 + 0.578i
Analytic conductor: 0.6647540.664754
Root analytic conductor: 0.8153240.815324
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1332(1027,⋅)\chi_{1332} (1027, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1332, ( :0), 0.815+0.578i)(2,\ 1332,\ (\ :0),\ 0.815 + 0.578i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9992251991.999225199
L(12)L(\frac12) ≈\approx 1.9992251991.999225199
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+(−0.984+0.173i)T 1 + (-0.984 + 0.173i)T
3 1 1
37 1+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
good5 1+(−0.223+0.266i)T+(−0.173−0.984i)T2 1 + (-0.223 + 0.266i)T + (-0.173 - 0.984i)T^{2}
7 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(0.592+1.62i)T+(−0.766+0.642i)T2 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2}
17 1+(0.524−1.43i)T+(−0.766−0.642i)T2 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2}
19 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
23 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1+(1.62−0.939i)T+(0.5−0.866i)T2 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2}
31 1+T2 1 + T^{2}
41 1+(−1.20+0.439i)T+(0.766−0.642i)T2 1 + (-1.20 + 0.439i)T + (0.766 - 0.642i)T^{2}
43 1+T2 1 + T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(1.32−1.11i)T+(0.173−0.984i)T2 1 + (1.32 - 1.11i)T + (0.173 - 0.984i)T^{2}
59 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
61 1+(0.233+0.642i)T+(−0.766+0.642i)T2 1 + (0.233 + 0.642i)T + (-0.766 + 0.642i)T^{2}
67 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
71 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
73 1+T+T2 1 + T + T^{2}
79 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
83 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
89 1+(0.984+1.17i)T+(−0.173+0.984i)T2 1 + (0.984 + 1.17i)T + (-0.173 + 0.984i)T^{2}
97 1+(0.592+0.342i)T+(0.5+0.866i)T2 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)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

−9.904954013899411001754664175730, −9.024052553641248413471286048354, −7.85082301808911264453205433688, −7.31185058835136623526987538299, −6.04020769334551485298489728345, −5.61941314543059012488020025323, −4.66301244424118899099197730439, −3.68261977244478790353354280985, −2.74101537663210139117901299788, −1.52962721299531859093282589190, 2.01220736707903967828861421397, 2.77959075112200319137318246810, 4.11679521970029403666211294793, 4.68724811624372878095831495828, 5.72629960985783079780850624330, 6.62649344902658929859756363844, 7.15047965132912870438780036123, 8.032015971377623711747054819289, 9.261816434328694850023014371229, 9.803799589969373007668376549092

Graph of the ZZ-function along the critical line