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Properties

Label 2-525-105.104-c1-0-22
Degree 22
Conductor 525525
Sign 0.968+0.247i0.968 + 0.247i
Analytic cond. 4.192144.19214
Root an. cond. 2.047472.04747
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + 1.73·3-s + 0.999·4-s − 2.99·6-s + (1.73 − 2i)7-s + 1.73·8-s + 2.99·9-s + 3.46i·11-s + 1.73·12-s + (−2.99 + 3.46i)14-s − 5·16-s − 6i·17-s − 5.19·18-s + 3.46i·19-s + (2.99 − 3.46i)21-s − 5.99i·22-s + ⋯
L(s)  = 1  − 1.22·2-s + 1.00·3-s + 0.499·4-s − 1.22·6-s + (0.654 − 0.755i)7-s + 0.612·8-s + 0.999·9-s + 1.04i·11-s + 0.500·12-s + (−0.801 + 0.925i)14-s − 1.25·16-s − 1.45i·17-s − 1.22·18-s + 0.794i·19-s + (0.654 − 0.755i)21-s − 1.27i·22-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((0.968+0.247i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=((0.968+0.247i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 0.968+0.247i0.968 + 0.247i
Analytic conductor: 4.192144.19214
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ525(524,⋅)\chi_{525} (524, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :1/2), 0.968+0.247i)(2,\ 525,\ (\ :1/2),\ 0.968 + 0.247i)

Particular Values

L(1)L(1) ≈\approx 1.15773−0.145520i1.15773 - 0.145520i
L(12)L(\frac12) ≈\approx 1.15773−0.145520i1.15773 - 0.145520i
L(32)L(\frac{3}{2}) 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 1−1.73T 1 - 1.73T
5 1 1
7 1+(−1.73+2i)T 1 + (-1.73 + 2i)T
good2 1+1.73T+2T2 1 + 1.73T + 2T^{2}
11 1−3.46iT−11T2 1 - 3.46iT - 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+6iT−17T2 1 + 6iT - 17T^{2}
19 1−3.46iT−19T2 1 - 3.46iT - 19T^{2}
23 1−3.46T+23T2 1 - 3.46T + 23T^{2}
29 1+6.92iT−29T2 1 + 6.92iT - 29T^{2}
31 1−3.46iT−31T2 1 - 3.46iT - 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1+8iT−43T2 1 + 8iT - 43T^{2}
47 1−12iT−47T2 1 - 12iT - 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1−12T+59T2 1 - 12T + 59T^{2}
61 1−6.92iT−61T2 1 - 6.92iT - 61T^{2}
67 1+8iT−67T2 1 + 8iT - 67T^{2}
71 1+3.46iT−71T2 1 + 3.46iT - 71T^{2}
73 1−6.92T+73T2 1 - 6.92T + 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+6.92T+97T2 1 + 6.92T + 97T^{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

−10.40335084660920893921396217859, −9.770338957526745238007914661872, −9.128940917908522275627797698986, −8.148946026181638605974642358321, −7.50528039832052153106196428743, −6.95027573315165821563669462409, −4.89855763857709975144732793745, −4.07421525887621953210749049990, −2.42076348425047111622674158741, −1.18870888224860901074469397589, 1.34277948835798500127276589344, 2.57606205125438410781092667339, 3.97900958360396786414068742996, 5.27944241422446973205677357206, 6.70267301208308914027839350482, 7.76705470977258941458935466993, 8.553375506640330609193478756077, 8.800401873638799884185958784555, 9.734371329967574109255276923591, 10.74468419557150499902276473547

Graph of the ZZ-function along the critical line