Properties

Label 2-425-85.19-c1-0-5
Degree 22
Conductor 425425
Sign 0.04480.998i0.0448 - 0.998i
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  + (−0.982 + 0.982i)2-s + (0.0424 + 0.102i)3-s + 0.0682i·4-s + (−0.142 − 0.0589i)6-s + (1.58 + 0.656i)7-s + (−2.03 − 2.03i)8-s + (2.11 − 2.11i)9-s + (5.35 + 2.21i)11-s + (−0.00698 + 0.00289i)12-s − 1.25·13-s + (−2.20 + 0.912i)14-s + 3.85·16-s + (1.85 + 3.68i)17-s + 4.15i·18-s + (−1.99 − 1.99i)19-s + ⋯
L(s)  = 1  + (−0.694 + 0.694i)2-s + (0.0244 + 0.0591i)3-s + 0.0341i·4-s + (−0.0580 − 0.0240i)6-s + (0.599 + 0.248i)7-s + (−0.718 − 0.718i)8-s + (0.704 − 0.704i)9-s + (1.61 + 0.669i)11-s + (−0.00201 + 0.000835i)12-s − 0.347·13-s + (−0.588 + 0.243i)14-s + 0.964·16-s + (0.448 + 0.893i)17-s + 0.978i·18-s + (−0.458 − 0.458i)19-s + ⋯

Functional equation

Λ(s)=(425s/2ΓC(s)L(s)=((0.04480.998i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0448 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2ΓC(s+1/2)L(s)=((0.04480.998i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0448 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52175^{2} \cdot 17
Sign: 0.04480.998i0.0448 - 0.998i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(274,)\chi_{425} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), 0.04480.998i)(2,\ 425,\ (\ :1/2),\ 0.0448 - 0.998i)

Particular Values

L(1)L(1) \approx 0.797089+0.762107i0.797089 + 0.762107i
L(12)L(\frac12) \approx 0.797089+0.762107i0.797089 + 0.762107i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
17 1+(1.853.68i)T 1 + (-1.85 - 3.68i)T
good2 1+(0.9820.982i)T2iT2 1 + (0.982 - 0.982i)T - 2iT^{2}
3 1+(0.04240.102i)T+(2.12+2.12i)T2 1 + (-0.0424 - 0.102i)T + (-2.12 + 2.12i)T^{2}
7 1+(1.580.656i)T+(4.94+4.94i)T2 1 + (-1.58 - 0.656i)T + (4.94 + 4.94i)T^{2}
11 1+(5.352.21i)T+(7.77+7.77i)T2 1 + (-5.35 - 2.21i)T + (7.77 + 7.77i)T^{2}
13 1+1.25T+13T2 1 + 1.25T + 13T^{2}
19 1+(1.99+1.99i)T+19iT2 1 + (1.99 + 1.99i)T + 19iT^{2}
23 1+(0.7851.89i)T+(16.216.2i)T2 1 + (0.785 - 1.89i)T + (-16.2 - 16.2i)T^{2}
29 1+(1.994.80i)T+(20.5+20.5i)T2 1 + (-1.99 - 4.80i)T + (-20.5 + 20.5i)T^{2}
31 1+(2.641.09i)T+(21.921.9i)T2 1 + (2.64 - 1.09i)T + (21.9 - 21.9i)T^{2}
37 1+(2.41+5.82i)T+(26.1+26.1i)T2 1 + (2.41 + 5.82i)T + (-26.1 + 26.1i)T^{2}
41 1+(3.618.73i)T+(28.928.9i)T2 1 + (3.61 - 8.73i)T + (-28.9 - 28.9i)T^{2}
43 1+(5.255.25i)T+43iT2 1 + (-5.25 - 5.25i)T + 43iT^{2}
47 17.63T+47T2 1 - 7.63T + 47T^{2}
53 1+(5.095.09i)T53iT2 1 + (5.09 - 5.09i)T - 53iT^{2}
59 1+(1.54+1.54i)T59iT2 1 + (-1.54 + 1.54i)T - 59iT^{2}
61 1+(2.19+5.30i)T+(43.143.1i)T2 1 + (-2.19 + 5.30i)T + (-43.1 - 43.1i)T^{2}
67 1+2.46iT67T2 1 + 2.46iT - 67T^{2}
71 1+(12.45.15i)T+(50.250.2i)T2 1 + (12.4 - 5.15i)T + (50.2 - 50.2i)T^{2}
73 1+(5.39+2.23i)T+(51.651.6i)T2 1 + (-5.39 + 2.23i)T + (51.6 - 51.6i)T^{2}
79 1+(3.621.50i)T+(55.8+55.8i)T2 1 + (-3.62 - 1.50i)T + (55.8 + 55.8i)T^{2}
83 1+(6.29+6.29i)T83iT2 1 + (-6.29 + 6.29i)T - 83iT^{2}
89 1+14.3iT89T2 1 + 14.3iT - 89T^{2}
97 1+(7.84+3.25i)T+(68.568.5i)T2 1 + (-7.84 + 3.25i)T + (68.5 - 68.5i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.47070401021883123675933044917, −10.16871863943241344725115194064, −9.299489056518308396000233491410, −8.779931267516346143849495517737, −7.65839600630523903854211449907, −6.88382863491993190552044933912, −6.11643551190516452674896671438, −4.50980382676382356940233395076, −3.53759462521202888489081660687, −1.48935913314382377239200469632, 1.07384121665871736601089773931, 2.23241237257769585860035452737, 3.87728965970461800645469058978, 5.08778758650533118669309878371, 6.28753447358901992636463595071, 7.44652506087874088937660780034, 8.458709187530877959450651978232, 9.267178349181861730817730444314, 10.14613457719529235222512777659, 10.83812841809820284586471122661

Graph of the ZZ-function along the critical line