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Properties

Label 2-425-17.2-c1-0-15
Degree 22
Conductor 425425
Sign −0.675+0.737i-0.675 + 0.737i
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  + (−1.82 − 1.82i)2-s + (2.84 − 1.17i)3-s + 4.67i·4-s + (−7.35 − 3.04i)6-s + (1.07 − 2.60i)7-s + (4.89 − 4.89i)8-s + (4.59 − 4.59i)9-s + (0.616 + 0.255i)11-s + (5.51 + 13.3i)12-s + 4.34i·13-s + (−6.72 + 2.78i)14-s − 8.52·16-s + (2.64 − 3.16i)17-s − 16.7·18-s + (1.88 + 1.88i)19-s + ⋯
L(s)  = 1  + (−1.29 − 1.29i)2-s + (1.64 − 0.680i)3-s + 2.33i·4-s + (−3.00 − 1.24i)6-s + (0.407 − 0.983i)7-s + (1.72 − 1.72i)8-s + (1.53 − 1.53i)9-s + (0.185 + 0.0769i)11-s + (1.59 + 3.84i)12-s + 1.20i·13-s + (−1.79 + 0.744i)14-s − 2.13·16-s + (0.640 − 0.767i)17-s − 3.95·18-s + (0.433 + 0.433i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.675+0.737i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.675+0.737i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.675+0.737i-0.675 + 0.737i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(376,⋅)\chi_{425} (376, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), −0.675+0.737i)(2,\ 425,\ (\ :1/2),\ -0.675 + 0.737i)

Particular Values

L(1)L(1) ≈\approx 0.528654−1.20145i0.528654 - 1.20145i
L(12)L(\frac12) ≈\approx 0.528654−1.20145i0.528654 - 1.20145i
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+(−2.64+3.16i)T 1 + (-2.64 + 3.16i)T
good2 1+(1.82+1.82i)T+2iT2 1 + (1.82 + 1.82i)T + 2iT^{2}
3 1+(−2.84+1.17i)T+(2.12−2.12i)T2 1 + (-2.84 + 1.17i)T + (2.12 - 2.12i)T^{2}
7 1+(−1.07+2.60i)T+(−4.94−4.94i)T2 1 + (-1.07 + 2.60i)T + (-4.94 - 4.94i)T^{2}
11 1+(−0.616−0.255i)T+(7.77+7.77i)T2 1 + (-0.616 - 0.255i)T + (7.77 + 7.77i)T^{2}
13 1−4.34iT−13T2 1 - 4.34iT - 13T^{2}
19 1+(−1.88−1.88i)T+19iT2 1 + (-1.88 - 1.88i)T + 19iT^{2}
23 1+(2.78+1.15i)T+(16.2+16.2i)T2 1 + (2.78 + 1.15i)T + (16.2 + 16.2i)T^{2}
29 1+(1.50+3.63i)T+(−20.5+20.5i)T2 1 + (1.50 + 3.63i)T + (-20.5 + 20.5i)T^{2}
31 1+(6.05−2.50i)T+(21.9−21.9i)T2 1 + (6.05 - 2.50i)T + (21.9 - 21.9i)T^{2}
37 1+(2.82−1.16i)T+(26.1−26.1i)T2 1 + (2.82 - 1.16i)T + (26.1 - 26.1i)T^{2}
41 1+(2.38−5.76i)T+(−28.9−28.9i)T2 1 + (2.38 - 5.76i)T + (-28.9 - 28.9i)T^{2}
43 1+(−4.37+4.37i)T−43iT2 1 + (-4.37 + 4.37i)T - 43iT^{2}
47 1+1.08iT−47T2 1 + 1.08iT - 47T^{2}
53 1+(4.94+4.94i)T+53iT2 1 + (4.94 + 4.94i)T + 53iT^{2}
59 1+(0.272−0.272i)T−59iT2 1 + (0.272 - 0.272i)T - 59iT^{2}
61 1+(4.24−10.2i)T+(−43.1−43.1i)T2 1 + (4.24 - 10.2i)T + (-43.1 - 43.1i)T^{2}
67 1+12.0T+67T2 1 + 12.0T + 67T^{2}
71 1+(−4.19+1.73i)T+(50.2−50.2i)T2 1 + (-4.19 + 1.73i)T + (50.2 - 50.2i)T^{2}
73 1+(−3.08−7.43i)T+(−51.6+51.6i)T2 1 + (-3.08 - 7.43i)T + (-51.6 + 51.6i)T^{2}
79 1+(−6.31−2.61i)T+(55.8+55.8i)T2 1 + (-6.31 - 2.61i)T + (55.8 + 55.8i)T^{2}
83 1+(−10.1−10.1i)T+83iT2 1 + (-10.1 - 10.1i)T + 83iT^{2}
89 1+0.844iT−89T2 1 + 0.844iT - 89T^{2}
97 1+(−4.46−10.7i)T+(−68.5+68.5i)T2 1 + (-4.46 - 10.7i)T + (-68.5 + 68.5i)T^{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

−10.62966853671169717802831385838, −9.668702122519307586690927687184, −9.205484833912586819301868855535, −8.260832402779264327490635857877, −7.59081766173963385650954927334, −6.97042946762103618964944383065, −4.15014978202593474947189452783, −3.34797909526048758706682886503, −2.13543984578405140712748433605, −1.24989077912236684490144122160, 1.87355188280708928900499971725, 3.37574913659667487736230622975, 5.06393877440666424331129085796, 5.95762309619458057423200703943, 7.55914332236753176383529720557, 7.912799199672709807933022699425, 8.836914821775752174432864865199, 9.203216738726132978379532524091, 10.11448339685003879497124021618, 10.84533484855384456090740744846

Graph of the ZZ-function along the critical line