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Properties

Label 2-425-17.2-c1-0-21
Degree 22
Conductor 425425
Sign 0.981+0.189i0.981 + 0.189i
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  + (0.813 + 0.813i)2-s + (1.89 − 0.786i)3-s − 0.677i·4-s + (2.18 + 0.904i)6-s + (0.143 − 0.346i)7-s + (2.17 − 2.17i)8-s + (0.866 − 0.866i)9-s + (0.0511 + 0.0211i)11-s + (−0.532 − 1.28i)12-s + 0.388i·13-s + (0.398 − 0.165i)14-s + 2.18·16-s + (−4.08 − 0.594i)17-s + 1.40·18-s + (1.25 + 1.25i)19-s + ⋯
L(s)  = 1  + (0.575 + 0.575i)2-s + (1.09 − 0.454i)3-s − 0.338i·4-s + (0.891 + 0.369i)6-s + (0.0543 − 0.131i)7-s + (0.769 − 0.769i)8-s + (0.288 − 0.288i)9-s + (0.0154 + 0.00639i)11-s + (−0.153 − 0.371i)12-s + 0.107i·13-s + (0.106 − 0.0441i)14-s + 0.546·16-s + (−0.989 − 0.144i)17-s + 0.332·18-s + (0.287 + 0.287i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.981+0.189i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((0.981+0.189i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.981+0.189i0.981 + 0.189i
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), 0.981+0.189i)(2,\ 425,\ (\ :1/2),\ 0.981 + 0.189i)

Particular Values

L(1)L(1) ≈\approx 2.57171−0.246532i2.57171 - 0.246532i
L(12)L(\frac12) ≈\approx 2.57171−0.246532i2.57171 - 0.246532i
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+(4.08+0.594i)T 1 + (4.08 + 0.594i)T
good2 1+(−0.813−0.813i)T+2iT2 1 + (-0.813 - 0.813i)T + 2iT^{2}
3 1+(−1.89+0.786i)T+(2.12−2.12i)T2 1 + (-1.89 + 0.786i)T + (2.12 - 2.12i)T^{2}
7 1+(−0.143+0.346i)T+(−4.94−4.94i)T2 1 + (-0.143 + 0.346i)T + (-4.94 - 4.94i)T^{2}
11 1+(−0.0511−0.0211i)T+(7.77+7.77i)T2 1 + (-0.0511 - 0.0211i)T + (7.77 + 7.77i)T^{2}
13 1−0.388iT−13T2 1 - 0.388iT - 13T^{2}
19 1+(−1.25−1.25i)T+19iT2 1 + (-1.25 - 1.25i)T + 19iT^{2}
23 1+(−0.948−0.392i)T+(16.2+16.2i)T2 1 + (-0.948 - 0.392i)T + (16.2 + 16.2i)T^{2}
29 1+(−3.23−7.80i)T+(−20.5+20.5i)T2 1 + (-3.23 - 7.80i)T + (-20.5 + 20.5i)T^{2}
31 1+(−5.59+2.31i)T+(21.9−21.9i)T2 1 + (-5.59 + 2.31i)T + (21.9 - 21.9i)T^{2}
37 1+(8.77−3.63i)T+(26.1−26.1i)T2 1 + (8.77 - 3.63i)T + (26.1 - 26.1i)T^{2}
41 1+(−2.93+7.08i)T+(−28.9−28.9i)T2 1 + (-2.93 + 7.08i)T + (-28.9 - 28.9i)T^{2}
43 1+(4.77−4.77i)T−43iT2 1 + (4.77 - 4.77i)T - 43iT^{2}
47 1+1.84iT−47T2 1 + 1.84iT - 47T^{2}
53 1+(6.43+6.43i)T+53iT2 1 + (6.43 + 6.43i)T + 53iT^{2}
59 1+(9.61−9.61i)T−59iT2 1 + (9.61 - 9.61i)T - 59iT^{2}
61 1+(2.22−5.36i)T+(−43.1−43.1i)T2 1 + (2.22 - 5.36i)T + (-43.1 - 43.1i)T^{2}
67 1−14.4T+67T2 1 - 14.4T + 67T^{2}
71 1+(1.04−0.430i)T+(50.2−50.2i)T2 1 + (1.04 - 0.430i)T + (50.2 - 50.2i)T^{2}
73 1+(−0.838−2.02i)T+(−51.6+51.6i)T2 1 + (-0.838 - 2.02i)T + (-51.6 + 51.6i)T^{2}
79 1+(3.64+1.50i)T+(55.8+55.8i)T2 1 + (3.64 + 1.50i)T + (55.8 + 55.8i)T^{2}
83 1+(6.67+6.67i)T+83iT2 1 + (6.67 + 6.67i)T + 83iT^{2}
89 1−2.61iT−89T2 1 - 2.61iT - 89T^{2}
97 1+(2.01+4.85i)T+(−68.5+68.5i)T2 1 + (2.01 + 4.85i)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

−11.08677137341586041733196629094, −10.17738173047988900781600477230, −9.132993080197832337846846142379, −8.330028800296717865412940522454, −7.27288774008674148911779260335, −6.63169072313655730863751743001, −5.38991975171489226893034950266, −4.35067557353121339739822789792, −3.04729406460627786292373446954, −1.60991926625598015347227448754, 2.21549020727799240269326217034, 3.09891805997689678098532492543, 4.06118160169683352131669175759, 4.95609822308764486165912723614, 6.52620224770816765602865735906, 7.84712163122586239106768569043, 8.493178924467650782278855873502, 9.334855909157935462848305150201, 10.36303571760696286380053339133, 11.32473298739174924628294415275

Graph of the ZZ-function along the critical line