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Properties

Label 2-2664-296.117-c0-0-0
Degree 22
Conductor 26642664
Sign 0.763−0.646i0.763 - 0.646i
Analytic cond. 1.329501.32950
Root an. cond. 1.153041.15304
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1 + i)5-s − 7-s + (−0.707 − 0.707i)8-s + 1.41i·10-s + 11-s + (−0.707 + 0.707i)13-s + (−0.707 + 0.707i)14-s − 1.00·16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)19-s + (1.00 + 1.00i)20-s + (0.707 − 0.707i)22-s + (0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1 + i)5-s − 7-s + (−0.707 − 0.707i)8-s + 1.41i·10-s + 11-s + (−0.707 + 0.707i)13-s + (−0.707 + 0.707i)14-s − 1.00·16-s + (0.707 + 0.707i)17-s + (−0.707 + 0.707i)19-s + (1.00 + 1.00i)20-s + (0.707 − 0.707i)22-s + (0.707 + 0.707i)23-s + ⋯

Functional equation

Λ(s)=(2664s/2 ΓC(s) L(s)=((0.763−0.646i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2664s/2 ΓC(s) L(s)=((0.763−0.646i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26642664    =    23⋅32⋅372^{3} \cdot 3^{2} \cdot 37
Sign: 0.763−0.646i0.763 - 0.646i
Analytic conductor: 1.329501.32950
Root analytic conductor: 1.153041.15304
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2664(2485,⋅)\chi_{2664} (2485, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2664, ( :0), 0.763−0.646i)(2,\ 2664,\ (\ :0),\ 0.763 - 0.646i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0459195901.045919590
L(12)L(\frac12) ≈\approx 1.0459195901.045919590
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
3 1 1
37 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good5 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
7 1+T+T2 1 + T + T^{2}
11 1−T+T2 1 - T + T^{2}
13 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
17 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
19 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
23 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
29 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
31 1−iT2 1 - iT^{2}
41 1−T2 1 - T^{2}
43 1−iT2 1 - iT^{2}
47 1+1.41T+T2 1 + 1.41T + T^{2}
53 1−iT−T2 1 - iT - T^{2}
59 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
61 1+(1.41+1.41i)T+iT2 1 + (1.41 + 1.41i)T + iT^{2}
67 1+1.41T+T2 1 + 1.41T + T^{2}
71 1−1.41T+T2 1 - 1.41T + T^{2}
73 1−iT−T2 1 - iT - T^{2}
79 1+iT2 1 + iT^{2}
83 1−iT−T2 1 - iT - T^{2}
89 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
97 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{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

−9.472559713564156631207106183397, −8.410998577678810418615313007699, −7.36535524830670398608775862261, −6.52733495964754336849084250535, −6.34361854443983488670577414989, −5.00742235302265401360787886456, −4.04723334864038131897076291148, −3.47193780973884609096448516902, −2.86383398815634168972815678434, −1.50882746121846903581356758917, 0.54988187467455456925364592165, 2.69673579205513706652654966853, 3.46609488298536598015911113412, 4.43459390172165722077128311553, 4.82184871844349074808866633567, 5.91041284891705055915272880482, 6.64816779659899490497414897329, 7.36890426413933048865381406351, 8.077263844721136873787702657043, 8.822113518880359493705864604439

Graph of the ZZ-function along the critical line