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Properties

Label 2-525-5.4-c3-0-8
Degree 22
Conductor 525525
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 30.976030.9760
Root an. cond. 5.565605.56560
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.23i·2-s + 3i·3-s − 9.94·4-s − 12.7·6-s + 7i·7-s − 8.23i·8-s − 9·9-s − 41.5·11-s − 29.8i·12-s + 88.9i·13-s − 29.6·14-s − 44.6·16-s + 120. i·17-s − 38.1i·18-s + 112.·19-s + ⋯
L(s)  = 1  + 1.49i·2-s + 0.577i·3-s − 1.24·4-s − 0.864·6-s + 0.377i·7-s − 0.363i·8-s − 0.333·9-s − 1.13·11-s − 0.717i·12-s + 1.89i·13-s − 0.566·14-s − 0.697·16-s + 1.71i·17-s − 0.499i·18-s + 1.35·19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+3/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 30.976030.9760
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ525(274,⋅)\chi_{525} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :3/2), 0.447+0.894i)(2,\ 525,\ (\ :3/2),\ 0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 0.94329850330.9432985033
L(12)L(\frac12) ≈\approx 0.94329850330.9432985033
L(52)L(\frac{5}{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 1−3iT 1 - 3iT
5 1 1
7 1−7iT 1 - 7iT
good2 1−4.23iT−8T2 1 - 4.23iT - 8T^{2}
11 1+41.5T+1.33e3T2 1 + 41.5T + 1.33e3T^{2}
13 1−88.9iT−2.19e3T2 1 - 88.9iT - 2.19e3T^{2}
17 1−120.iT−4.91e3T2 1 - 120. iT - 4.91e3T^{2}
19 1−112.T+6.85e3T2 1 - 112.T + 6.85e3T^{2}
23 1+115.iT−1.21e4T2 1 + 115. iT - 1.21e4T^{2}
29 1−144.T+2.43e4T2 1 - 144.T + 2.43e4T^{2}
31 1+258.T+2.97e4T2 1 + 258.T + 2.97e4T^{2}
37 1+48.3iT−5.06e4T2 1 + 48.3iT - 5.06e4T^{2}
41 1−200.T+6.89e4T2 1 - 200.T + 6.89e4T^{2}
43 1+218.iT−7.95e4T2 1 + 218. iT - 7.95e4T^{2}
47 1+575.iT−1.03e5T2 1 + 575. iT - 1.03e5T^{2}
53 1+184.iT−1.48e5T2 1 + 184. iT - 1.48e5T^{2}
59 1−151.T+2.05e5T2 1 - 151.T + 2.05e5T^{2}
61 1+529.T+2.26e5T2 1 + 529.T + 2.26e5T^{2}
67 1+1.28iT−3.00e5T2 1 + 1.28iT - 3.00e5T^{2}
71 1+61.4T+3.57e5T2 1 + 61.4T + 3.57e5T^{2}
73 1−484.iT−3.89e5T2 1 - 484. iT - 3.89e5T^{2}
79 1+878.T+4.93e5T2 1 + 878.T + 4.93e5T^{2}
83 1−491.iT−5.71e5T2 1 - 491. iT - 5.71e5T^{2}
89 1−415.T+7.04e5T2 1 - 415.T + 7.04e5T^{2}
97 1−1.03e3iT−9.12e5T2 1 - 1.03e3iT - 9.12e5T^{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

−11.04569102467808904206876435311, −10.10178087538070155270015979860, −9.019983068034292439466931812387, −8.505025928829268358651699910681, −7.50474014685843443155313603229, −6.59731488510146135835771490082, −5.70485548349316412714677842163, −4.90853104921410795360070234082, −3.88900486469867854799395718820, −2.18498110468207537380224385567, 0.30721832406503930222481964623, 1.21100416868405288663222589763, 2.83528033427096463647666197159, 3.13253116104481272972588001508, 4.83976750361355594094720533691, 5.68003048887681946222963262919, 7.38996629216998087385113079209, 7.76712190599655738149455224128, 9.215197978436076251178106851999, 9.948035445078937996404182533189

Graph of the ZZ-function along the critical line