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Properties

Label 2-425-17.16-c1-0-25
Degree 22
Conductor 425425
Sign −0.765+0.643i-0.765 + 0.643i
Analytic cond. 3.393643.39364
Root an. cond. 1.842181.84218
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  + 1.48·2-s − 3.29i·3-s + 0.193·4-s − 4.87i·6-s − 2.22i·7-s − 2.67·8-s − 7.83·9-s + 2.86i·11-s − 0.638i·12-s + 2.28·13-s − 3.29i·14-s − 4.35·16-s + (3.15 − 2.65i)17-s − 11.5·18-s + 5.76·19-s + ⋯
L(s)  = 1  + 1.04·2-s − 1.90i·3-s + 0.0969·4-s − 1.99i·6-s − 0.839i·7-s − 0.945·8-s − 2.61·9-s + 0.862i·11-s − 0.184i·12-s + 0.634·13-s − 0.879i·14-s − 1.08·16-s + (0.765 − 0.643i)17-s − 2.73·18-s + 1.32·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.765+0.643i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.765+0.643i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.765+0.643i-0.765 + 0.643i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(101,⋅)\chi_{425} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), −0.765+0.643i)(2,\ 425,\ (\ :1/2),\ -0.765 + 0.643i)

Particular Values

L(1)L(1) ≈\approx 0.639088−1.75366i0.639088 - 1.75366i
L(12)L(\frac12) ≈\approx 0.639088−1.75366i0.639088 - 1.75366i
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)
bad5 1 1
17 1+(−3.15+2.65i)T 1 + (-3.15 + 2.65i)T
good2 1−1.48T+2T2 1 - 1.48T + 2T^{2}
3 1+3.29iT−3T2 1 + 3.29iT - 3T^{2}
7 1+2.22iT−7T2 1 + 2.22iT - 7T^{2}
11 1−2.86iT−11T2 1 - 2.86iT - 11T^{2}
13 1−2.28T+13T2 1 - 2.28T + 13T^{2}
19 1−5.76T+19T2 1 - 5.76T + 19T^{2}
23 1+1.58iT−23T2 1 + 1.58iT - 23T^{2}
29 1+9.23iT−29T2 1 + 9.23iT - 29T^{2}
31 1−1.15iT−31T2 1 - 1.15iT - 31T^{2}
37 1+0.514iT−37T2 1 + 0.514iT - 37T^{2}
41 1+7.09iT−41T2 1 + 7.09iT - 41T^{2}
43 1+7.89T+43T2 1 + 7.89T + 43T^{2}
47 1−3.03T+47T2 1 - 3.03T + 47T^{2}
53 1−5.73T+53T2 1 - 5.73T + 53T^{2}
59 1+7.50T+59T2 1 + 7.50T + 59T^{2}
61 1−11.8iT−61T2 1 - 11.8iT - 61T^{2}
67 1−7.35T+67T2 1 - 7.35T + 67T^{2}
71 1−8.80iT−71T2 1 - 8.80iT - 71T^{2}
73 1−2.65iT−73T2 1 - 2.65iT - 73T^{2}
79 1−11.4iT−79T2 1 - 11.4iT - 79T^{2}
83 1−3.08T+83T2 1 - 3.08T + 83T^{2}
89 1−2.15T+89T2 1 - 2.15T + 89T^{2}
97 1+8.72iT−97T2 1 + 8.72iT - 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

−11.47609297996199021195190339686, −9.963663646947963552144440333679, −8.744962564256339881819426112396, −7.64203025291849273891202041040, −7.06994747418436006377180009572, −6.08538601996168966526959932757, −5.20037599032290634139940591865, −3.75554655579716091157645102437, −2.52046630249875102085205311784, −0.912105174608685865190191561326, 3.23209841217578580637727543108, 3.46657527955908020102966059679, 4.81725843405412052491277975555, 5.47778936992801384962393316123, 6.10133655695673974867401004248, 8.284204554202767961512558465371, 9.001998240742008648035340390595, 9.672668978451985835844495248360, 10.73523495519081327458972201026, 11.52284607153626153384823842564

Graph of the ZZ-function along the critical line