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Properties

Label 2-575-5.4-c3-0-6
Degree 22
Conductor 575575
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 33.926033.9260
Root an. cond. 5.824615.82461
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  − 4.96i·2-s − 7.15i·3-s − 16.6·4-s − 35.5·6-s + 3.69i·7-s + 42.7i·8-s − 24.2·9-s − 25.2·11-s + 118. i·12-s + 29.3i·13-s + 18.3·14-s + 79.1·16-s + 80.9i·17-s + 120. i·18-s + 48.2·19-s + ⋯
L(s)  = 1  − 1.75i·2-s − 1.37i·3-s − 2.07·4-s − 2.41·6-s + 0.199i·7-s + 1.88i·8-s − 0.896·9-s − 0.693·11-s + 2.86i·12-s + 0.625i·13-s + 0.349·14-s + 1.23·16-s + 1.15i·17-s + 1.57i·18-s + 0.582·19-s + ⋯

Functional equation

Λ(s)=(575s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(575s/2 ΓC(s+3/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{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: 575575    =    52⋅235^{2} \cdot 23
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 33.926033.9260
Root analytic conductor: 5.824615.82461
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ575(24,⋅)\chi_{575} (24, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 575, ( :3/2), 0.447+0.894i)(2,\ 575,\ (\ :3/2),\ 0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 0.66563621440.6656362144
L(12)L(\frac12) ≈\approx 0.66563621440.6656362144
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)
bad5 1 1
23 1−23iT 1 - 23iT
good2 1+4.96iT−8T2 1 + 4.96iT - 8T^{2}
3 1+7.15iT−27T2 1 + 7.15iT - 27T^{2}
7 1−3.69iT−343T2 1 - 3.69iT - 343T^{2}
11 1+25.2T+1.33e3T2 1 + 25.2T + 1.33e3T^{2}
13 1−29.3iT−2.19e3T2 1 - 29.3iT - 2.19e3T^{2}
17 1−80.9iT−4.91e3T2 1 - 80.9iT - 4.91e3T^{2}
19 1−48.2T+6.85e3T2 1 - 48.2T + 6.85e3T^{2}
29 1−161.T+2.43e4T2 1 - 161.T + 2.43e4T^{2}
31 1+261.T+2.97e4T2 1 + 261.T + 2.97e4T^{2}
37 1+37.1iT−5.06e4T2 1 + 37.1iT - 5.06e4T^{2}
41 1+135.T+6.89e4T2 1 + 135.T + 6.89e4T^{2}
43 1−304.iT−7.95e4T2 1 - 304. iT - 7.95e4T^{2}
47 1−200.iT−1.03e5T2 1 - 200. iT - 1.03e5T^{2}
53 1+384.iT−1.48e5T2 1 + 384. iT - 1.48e5T^{2}
59 1+113.T+2.05e5T2 1 + 113.T + 2.05e5T^{2}
61 1+763.T+2.26e5T2 1 + 763.T + 2.26e5T^{2}
67 1−736.iT−3.00e5T2 1 - 736. iT - 3.00e5T^{2}
71 1−721.T+3.57e5T2 1 - 721.T + 3.57e5T^{2}
73 1+380.iT−3.89e5T2 1 + 380. iT - 3.89e5T^{2}
79 1+754.T+4.93e5T2 1 + 754.T + 4.93e5T^{2}
83 1−1.18e3iT−5.71e5T2 1 - 1.18e3iT - 5.71e5T^{2}
89 1+611.T+7.04e5T2 1 + 611.T + 7.04e5T^{2}
97 1+371.iT−9.12e5T2 1 + 371. iT - 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

−10.42052993829885563929255167562, −9.469211873940126044313536256686, −8.537285549005033450547886682533, −7.70716679459698732726672511399, −6.57943958333695508612400252045, −5.39100374557591419276929767613, −4.11021936565182095476877787367, −2.90722680952208905230533158755, −1.95749975864769479801103215298, −1.16487393682732829392293628168, 0.21999693834752914058389280658, 3.12845530579733731621139992706, 4.31190681461206794674446533583, 5.12930830513413430404681391330, 5.64405286598408971864614133597, 6.97679627404191382927944118205, 7.68875365203034957317047536212, 8.689210889859432384517035660276, 9.382268620349151378065105768773, 10.17850000694931834986363889509

Graph of the ZZ-function along the critical line