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Properties

Label 2-3724-76.11-c0-0-2
Degree 22
Conductor 37243724
Sign −0.211−0.977i-0.211 - 0.977i
Analytic cond. 1.858511.85851
Root an. cond. 1.363271.36327
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  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999i·8-s + i·11-s + 0.999i·12-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.5 + 0.866i)22-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + (0.5 + 0.866i)25-s + 0.999i·26-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999i·8-s + i·11-s + 0.999i·12-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.5 + 0.866i)22-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)24-s + (0.5 + 0.866i)25-s + 0.999i·26-s + ⋯

Functional equation

Λ(s)=(3724s/2 ΓC(s) L(s)=((−0.211−0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3724s/2 ΓC(s) L(s)=((−0.211−0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37243724    =    22⋅72⋅192^{2} \cdot 7^{2} \cdot 19
Sign: −0.211−0.977i-0.211 - 0.977i
Analytic conductor: 1.858511.85851
Root analytic conductor: 1.363271.36327
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3724(3431,⋅)\chi_{3724} (3431, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3724, ( :0), −0.211−0.977i)(2,\ 3724,\ (\ :0),\ -0.211 - 0.977i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.8153275522.815327552
L(12)L(\frac12) ≈\approx 2.8153275522.815327552
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
7 1 1
19 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
good3 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
5 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
11 1−iT−T2 1 - iT - T^{2}
13 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
29 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
31 1+iT−T2 1 + iT - T^{2}
37 1−T+T2 1 - T + T^{2}
41 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)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+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
59 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
73 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
97 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)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

−9.058237094233260455568179815225, −7.86906738510917640599563475639, −7.60795011787031128138602774869, −6.55317951424837402842115975678, −6.02189186281492031888359515432, −4.91036020281776479070697520718, −4.27026675870591517719560576562, −3.68553179156600485277215250780, −2.72956116339424830848870014805, −1.96731395285619897931016149609, 1.16019925302006036032582390610, 2.17293044368975333148887017350, 2.99915762846065234162644832217, 3.62466116095247499080432083322, 4.47272940884091267071342294289, 5.73484887567308191455206168245, 5.91989624513621390697092791723, 6.96323025341566582446910436811, 7.83955196063509924204949394575, 8.496607066947543617901941050601

Graph of the ZZ-function along the critical line