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Properties

Label 2-525-105.47-c0-0-0
Degree 22
Conductor 525525
Sign 0.864−0.503i0.864 - 0.503i
Analytic cond. 0.2620090.262009
Root an. cond. 0.5118680.511868
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  + (−0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + (0.965 + 0.258i)7-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)12-s + (−0.707 − 0.707i)13-s + (0.499 − 0.866i)16-s + (−0.866 + 1.5i)19-s + (−0.499 + 0.866i)21-s + (0.707 − 0.707i)27-s + (0.965 − 0.258i)28-s − 36-s + (−1.67 + 0.448i)37-s + (0.866 − 0.500i)39-s + (0.707 + 0.707i)48-s + (0.866 + 0.499i)49-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)3-s + (0.866 − 0.5i)4-s + (0.965 + 0.258i)7-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)12-s + (−0.707 − 0.707i)13-s + (0.499 − 0.866i)16-s + (−0.866 + 1.5i)19-s + (−0.499 + 0.866i)21-s + (0.707 − 0.707i)27-s + (0.965 − 0.258i)28-s − 36-s + (−1.67 + 0.448i)37-s + (0.866 − 0.500i)39-s + (0.707 + 0.707i)48-s + (0.866 + 0.499i)49-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((0.864−0.503i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s) L(s)=((0.864−0.503i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 0.864−0.503i0.864 - 0.503i
Analytic conductor: 0.2620090.262009
Root analytic conductor: 0.5118680.511868
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ525(257,⋅)\chi_{525} (257, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :0), 0.864−0.503i)(2,\ 525,\ (\ :0),\ 0.864 - 0.503i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0047594631.004759463
L(12)L(\frac12) ≈\approx 1.0047594631.004759463
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+(0.258−0.965i)T 1 + (0.258 - 0.965i)T
5 1 1
7 1+(−0.965−0.258i)T 1 + (-0.965 - 0.258i)T
good2 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
17 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
19 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
29 1+T2 1 + T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(1.67−0.448i)T+(0.866−0.5i)T2 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2}
41 1+T2 1 + T^{2}
43 1−iT2 1 - iT^{2}
47 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
53 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
67 1+(−0.448+1.67i)T+(−0.866−0.5i)T2 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2}
71 1−T2 1 - T^{2}
73 1+(−0.258+0.965i)T+(−0.866−0.5i)T2 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}
79 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
83 1−iT2 1 - iT^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)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

−10.85265194667458318473597460308, −10.49853657569453420316314641092, −9.646041102116797199558271163784, −8.479199127412655074424145371704, −7.64476829139186849407586067741, −6.33016992215885247529100964567, −5.49874314575324618887090186657, −4.71787999659229431119451120672, −3.31966094801932222018327337308, −1.93422821101657829483960274435, 1.75017213250817479610146388580, 2.68599220391535101980008274751, 4.37837114333564791038553300389, 5.54458117085940507784996483608, 6.86617906086763478105011728454, 7.12622957452280402084405631376, 8.159727468546776311303523826892, 8.911157681126444530990841185717, 10.51636380611249884633618555736, 11.20850773412826818690495518803

Graph of the ZZ-function along the critical line