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Properties

Label 2-2925-117.70-c0-0-0
Degree 22
Conductor 29252925
Sign 0.919+0.393i0.919 + 0.393i
Analytic cond. 1.459761.45976
Root an. cond. 1.208201.20820
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  − 3-s + (0.866 − 0.5i)4-s + 9-s + (1.36 − 0.366i)11-s + (−0.866 + 0.5i)12-s + (0.5 + 0.866i)13-s + (0.499 − 0.866i)16-s − i·17-s + (−1 + i)19-s + (0.866 − 0.5i)23-s − 27-s + (−0.5 + 0.866i)29-s + (−1.36 + 0.366i)33-s + (0.866 − 0.5i)36-s + (1 + i)37-s + ⋯
L(s)  = 1  − 3-s + (0.866 − 0.5i)4-s + 9-s + (1.36 − 0.366i)11-s + (−0.866 + 0.5i)12-s + (0.5 + 0.866i)13-s + (0.499 − 0.866i)16-s − i·17-s + (−1 + i)19-s + (0.866 − 0.5i)23-s − 27-s + (−0.5 + 0.866i)29-s + (−1.36 + 0.366i)33-s + (0.866 − 0.5i)36-s + (1 + i)37-s + ⋯

Functional equation

Λ(s)=(2925s/2 ΓC(s) L(s)=((0.919+0.393i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2925s/2 ΓC(s) L(s)=((0.919+0.393i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29252925    =    32⋅52⋅133^{2} \cdot 5^{2} \cdot 13
Sign: 0.919+0.393i0.919 + 0.393i
Analytic conductor: 1.459761.45976
Root analytic conductor: 1.208201.20820
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2925(2176,⋅)\chi_{2925} (2176, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2925, ( :0), 0.919+0.393i)(2,\ 2925,\ (\ :0),\ 0.919 + 0.393i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2509653461.250965346
L(12)L(\frac12) ≈\approx 1.2509653461.250965346
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)
bad3 1+T 1 + T
5 1 1
13 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good2 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
7 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
11 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
17 1+iT−T2 1 + iT - T^{2}
19 1+(1−i)T−iT2 1 + (1 - i)T - iT^{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+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
37 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
41 1+(1.36+0.366i)T+(0.866+0.5i)T2 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
53 1+T+T2 1 + T + T^{2}
59 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
67 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
71 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
73 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
79 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
89 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
97 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)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

−9.124409765291113233380648230946, −8.031784457276360488624593519187, −6.90511310109354573344025656714, −6.64466356839560913280319171622, −6.04125062300452721235416149510, −5.14988178273420134382604218256, −4.30076231456971782866763306704, −3.33878965528293960305401460200, −1.91019180928855719358095150549, −1.12622466143218779620103461745, 1.21463547169001404971004794987, 2.24060520681680874517357841782, 3.58328680220920211981000917737, 4.16985925833762368292470743684, 5.25638626374358938827516945752, 6.27178089563315163664028425376, 6.48223551884187920856939801801, 7.33515845797171779467760492579, 8.086654665716087683620495688198, 8.987257285186787705309664075447

Graph of the ZZ-function along the critical line