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Properties

Label 2-525-5.4-c5-0-40
Degree 22
Conductor 525525
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 84.201584.2015
Root an. cond. 9.176139.17613
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 9i·3-s + 31·4-s − 9·6-s − 49i·7-s + 63i·8-s − 81·9-s − 340·11-s + 279i·12-s − 454i·13-s + 49·14-s + 929·16-s − 798i·17-s − 81i·18-s − 892·19-s + ⋯
L(s)  = 1  + 0.176i·2-s + 0.577i·3-s + 0.968·4-s − 0.102·6-s − 0.377i·7-s + 0.348i·8-s − 0.333·9-s − 0.847·11-s + 0.559i·12-s − 0.745i·13-s + 0.0668·14-s + 0.907·16-s − 0.669i·17-s − 0.0589i·18-s − 0.566·19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+5/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 84.201584.2015
Root analytic conductor: 9.176139.17613
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ525(274,⋅)\chi_{525} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :5/2), 0.447−0.894i)(2,\ 525,\ (\ :5/2),\ 0.447 - 0.894i)

Particular Values

L(3)L(3) ≈\approx 2.5789641822.578964182
L(12)L(\frac12) ≈\approx 2.5789641822.578964182
L(72)L(\frac{7}{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 1−9iT 1 - 9iT
5 1 1
7 1+49iT 1 + 49iT
good2 1−iT−32T2 1 - iT - 32T^{2}
11 1+340T+1.61e5T2 1 + 340T + 1.61e5T^{2}
13 1+454iT−3.71e5T2 1 + 454iT - 3.71e5T^{2}
17 1+798iT−1.41e6T2 1 + 798iT - 1.41e6T^{2}
19 1+892T+2.47e6T2 1 + 892T + 2.47e6T^{2}
23 1−3.19e3iT−6.43e6T2 1 - 3.19e3iT - 6.43e6T^{2}
29 1−8.24e3T+2.05e7T2 1 - 8.24e3T + 2.05e7T^{2}
31 1+2.49e3T+2.86e7T2 1 + 2.49e3T + 2.86e7T^{2}
37 1−9.79e3iT−6.93e7T2 1 - 9.79e3iT - 6.93e7T^{2}
41 1−1.98e4T+1.15e8T2 1 - 1.98e4T + 1.15e8T^{2}
43 1−1.72e4iT−1.47e8T2 1 - 1.72e4iT - 1.47e8T^{2}
47 1−8.92e3iT−2.29e8T2 1 - 8.92e3iT - 2.29e8T^{2}
53 1+150iT−4.18e8T2 1 + 150iT - 4.18e8T^{2}
59 1−4.23e4T+7.14e8T2 1 - 4.23e4T + 7.14e8T^{2}
61 1−1.47e4T+8.44e8T2 1 - 1.47e4T + 8.44e8T^{2}
67 1+1.67e3iT−1.35e9T2 1 + 1.67e3iT - 1.35e9T^{2}
71 1−1.45e4T+1.80e9T2 1 - 1.45e4T + 1.80e9T^{2}
73 1+7.83e4iT−2.07e9T2 1 + 7.83e4iT - 2.07e9T^{2}
79 1−2.27e3T+3.07e9T2 1 - 2.27e3T + 3.07e9T^{2}
83 1−3.77e4iT−3.93e9T2 1 - 3.77e4iT - 3.93e9T^{2}
89 1−1.17e5T+5.58e9T2 1 - 1.17e5T + 5.58e9T^{2}
97 1−1.00e4iT−8.58e9T2 1 - 1.00e4iT - 8.58e9T^{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

−10.34855154126651163910116529711, −9.539930802938821547542978563384, −8.201232510173380377435456814628, −7.62301333734901828672932142925, −6.55510443540717000154773896784, −5.60513849393895086389154288704, −4.68372193317830467201781981675, −3.28352992719996725955775250561, −2.49337970870526630662315142631, −0.937123339118542102148558613541, 0.66932771822843891191417486954, 2.07527145332434108218637031657, 2.60346247328115620667975010029, 4.05388294609623167481252015840, 5.47555024839076003730230663124, 6.38965058811131032132936348056, 7.07192354727656082623212264657, 8.093359423733256485403281274681, 8.841570237401453904916397195269, 10.21877312442153472369271278421

Graph of the ZZ-function along the critical line