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Properties

Label 2-425-17.9-c1-0-12
Degree 22
Conductor 425425
Sign 0.913+0.406i0.913 + 0.406i
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 + (−0.102 − 0.0424i)3-s + 0.0682i·4-s + (−0.142 + 0.0589i)6-s + (0.656 + 1.58i)7-s + (2.03 + 2.03i)8-s + (−2.11 − 2.11i)9-s + (5.35 − 2.21i)11-s + (0.00289 − 0.00698i)12-s + 1.25i·13-s + (2.20 + 0.912i)14-s + 3.85·16-s + (3.68 + 1.85i)17-s − 4.15·18-s + (1.99 − 1.99i)19-s + ⋯
L(s)  = 1  + (0.694 − 0.694i)2-s + (−0.0591 − 0.0244i)3-s + 0.0341i·4-s + (−0.0580 + 0.0240i)6-s + (0.248 + 0.599i)7-s + (0.718 + 0.718i)8-s + (−0.704 − 0.704i)9-s + (1.61 − 0.669i)11-s + (0.000835 − 0.00201i)12-s + 0.347i·13-s + (0.588 + 0.243i)14-s + 0.964·16-s + (0.893 + 0.448i)17-s − 0.978·18-s + (0.458 − 0.458i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.913+0.406i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((0.913+0.406i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.913+0.406i0.913 + 0.406i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(26,⋅)\chi_{425} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), 0.913+0.406i)(2,\ 425,\ (\ :1/2),\ 0.913 + 0.406i)

Particular Values

L(1)L(1) ≈\approx 2.06126−0.438032i2.06126 - 0.438032i
L(12)L(\frac12) ≈\approx 2.06126−0.438032i2.06126 - 0.438032i
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+(−3.68−1.85i)T 1 + (-3.68 - 1.85i)T
good2 1+(−0.982+0.982i)T−2iT2 1 + (-0.982 + 0.982i)T - 2iT^{2}
3 1+(0.102+0.0424i)T+(2.12+2.12i)T2 1 + (0.102 + 0.0424i)T + (2.12 + 2.12i)T^{2}
7 1+(−0.656−1.58i)T+(−4.94+4.94i)T2 1 + (-0.656 - 1.58i)T + (-4.94 + 4.94i)T^{2}
11 1+(−5.35+2.21i)T+(7.77−7.77i)T2 1 + (-5.35 + 2.21i)T + (7.77 - 7.77i)T^{2}
13 1−1.25iT−13T2 1 - 1.25iT - 13T^{2}
19 1+(−1.99+1.99i)T−19iT2 1 + (-1.99 + 1.99i)T - 19iT^{2}
23 1+(1.89−0.785i)T+(16.2−16.2i)T2 1 + (1.89 - 0.785i)T + (16.2 - 16.2i)T^{2}
29 1+(1.99−4.80i)T+(−20.5−20.5i)T2 1 + (1.99 - 4.80i)T + (-20.5 - 20.5i)T^{2}
31 1+(2.64+1.09i)T+(21.9+21.9i)T2 1 + (2.64 + 1.09i)T + (21.9 + 21.9i)T^{2}
37 1+(5.82+2.41i)T+(26.1+26.1i)T2 1 + (5.82 + 2.41i)T + (26.1 + 26.1i)T^{2}
41 1+(3.61+8.73i)T+(−28.9+28.9i)T2 1 + (3.61 + 8.73i)T + (-28.9 + 28.9i)T^{2}
43 1+(5.25+5.25i)T+43iT2 1 + (5.25 + 5.25i)T + 43iT^{2}
47 1−7.63iT−47T2 1 - 7.63iT - 47T^{2}
53 1+(5.09−5.09i)T−53iT2 1 + (5.09 - 5.09i)T - 53iT^{2}
59 1+(1.54+1.54i)T+59iT2 1 + (1.54 + 1.54i)T + 59iT^{2}
61 1+(−2.19−5.30i)T+(−43.1+43.1i)T2 1 + (-2.19 - 5.30i)T + (-43.1 + 43.1i)T^{2}
67 1+2.46T+67T2 1 + 2.46T + 67T^{2}
71 1+(12.4+5.15i)T+(50.2+50.2i)T2 1 + (12.4 + 5.15i)T + (50.2 + 50.2i)T^{2}
73 1+(−2.23+5.39i)T+(−51.6−51.6i)T2 1 + (-2.23 + 5.39i)T + (-51.6 - 51.6i)T^{2}
79 1+(3.62−1.50i)T+(55.8−55.8i)T2 1 + (3.62 - 1.50i)T + (55.8 - 55.8i)T^{2}
83 1+(−6.29+6.29i)T−83iT2 1 + (-6.29 + 6.29i)T - 83iT^{2}
89 1+14.3iT−89T2 1 + 14.3iT - 89T^{2}
97 1+(3.25−7.84i)T+(−68.5−68.5i)T2 1 + (3.25 - 7.84i)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

−11.57408326804240952239653593835, −10.54760876179308192454892279153, −9.090566115220771815689893975230, −8.709103591773664242067351928034, −7.36900055443559678251220418347, −6.13811972544696826548412513883, −5.26931089124875677267594557513, −3.85423199802912131244405605488, −3.22538781940504893322124686988, −1.64813817300117692692136875802, 1.48060230022180798708184310949, 3.54136763117212357938297153639, 4.59293462324323925370854382630, 5.47896994859908116063979625231, 6.45737623853928105462610579380, 7.34145155743133988672458556009, 8.203888587807946623158810719141, 9.610110432044029100312621994051, 10.23065013605488505903924246786, 11.37996282112270668370778229232

Graph of the ZZ-function along the critical line