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Properties

Label 2-425-17.2-c1-0-13
Degree 22
Conductor 425425
Sign −0.209+0.977i-0.209 + 0.977i
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  + (0.392 + 0.392i)2-s + (−2.31 + 0.958i)3-s − 1.69i·4-s + (−1.28 − 0.532i)6-s + (−1.54 + 3.72i)7-s + (1.45 − 1.45i)8-s + (2.31 − 2.31i)9-s + (−0.388 − 0.160i)11-s + (1.62 + 3.91i)12-s − 4.96i·13-s + (−2.07 + 0.858i)14-s − 2.24·16-s + (0.676 − 4.06i)17-s + 1.81·18-s + (−3.56 − 3.56i)19-s + ⋯
L(s)  = 1  + (0.277 + 0.277i)2-s + (−1.33 + 0.553i)3-s − 0.845i·4-s + (−0.524 − 0.217i)6-s + (−0.583 + 1.40i)7-s + (0.512 − 0.512i)8-s + (0.770 − 0.770i)9-s + (−0.117 − 0.0485i)11-s + (0.467 + 1.12i)12-s − 1.37i·13-s + (−0.553 + 0.229i)14-s − 0.560·16-s + (0.164 − 0.986i)17-s + 0.427·18-s + (−0.817 − 0.817i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.209+0.977i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.209+0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.209+0.977i-0.209 + 0.977i
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), −0.209+0.977i)(2,\ 425,\ (\ :1/2),\ -0.209 + 0.977i)

Particular Values

L(1)L(1) ≈\approx 0.277415−0.343230i0.277415 - 0.343230i
L(12)L(\frac12) ≈\approx 0.277415−0.343230i0.277415 - 0.343230i
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+(−0.676+4.06i)T 1 + (-0.676 + 4.06i)T
good2 1+(−0.392−0.392i)T+2iT2 1 + (-0.392 - 0.392i)T + 2iT^{2}
3 1+(2.31−0.958i)T+(2.12−2.12i)T2 1 + (2.31 - 0.958i)T + (2.12 - 2.12i)T^{2}
7 1+(1.54−3.72i)T+(−4.94−4.94i)T2 1 + (1.54 - 3.72i)T + (-4.94 - 4.94i)T^{2}
11 1+(0.388+0.160i)T+(7.77+7.77i)T2 1 + (0.388 + 0.160i)T + (7.77 + 7.77i)T^{2}
13 1+4.96iT−13T2 1 + 4.96iT - 13T^{2}
19 1+(3.56+3.56i)T+19iT2 1 + (3.56 + 3.56i)T + 19iT^{2}
23 1+(7.44+3.08i)T+(16.2+16.2i)T2 1 + (7.44 + 3.08i)T + (16.2 + 16.2i)T^{2}
29 1+(0.0710+0.171i)T+(−20.5+20.5i)T2 1 + (0.0710 + 0.171i)T + (-20.5 + 20.5i)T^{2}
31 1+(−3.15+1.30i)T+(21.9−21.9i)T2 1 + (-3.15 + 1.30i)T + (21.9 - 21.9i)T^{2}
37 1+(−3.52+1.46i)T+(26.1−26.1i)T2 1 + (-3.52 + 1.46i)T + (26.1 - 26.1i)T^{2}
41 1+(1.49−3.61i)T+(−28.9−28.9i)T2 1 + (1.49 - 3.61i)T + (-28.9 - 28.9i)T^{2}
43 1+(3.64−3.64i)T−43iT2 1 + (3.64 - 3.64i)T - 43iT^{2}
47 1−6.56iT−47T2 1 - 6.56iT - 47T^{2}
53 1+(0.616+0.616i)T+53iT2 1 + (0.616 + 0.616i)T + 53iT^{2}
59 1+(−10.1+10.1i)T−59iT2 1 + (-10.1 + 10.1i)T - 59iT^{2}
61 1+(−1.30+3.15i)T+(−43.1−43.1i)T2 1 + (-1.30 + 3.15i)T + (-43.1 - 43.1i)T^{2}
67 1+7.21T+67T2 1 + 7.21T + 67T^{2}
71 1+(6.91−2.86i)T+(50.2−50.2i)T2 1 + (6.91 - 2.86i)T + (50.2 - 50.2i)T^{2}
73 1+(1.37+3.31i)T+(−51.6+51.6i)T2 1 + (1.37 + 3.31i)T + (-51.6 + 51.6i)T^{2}
79 1+(13.0+5.38i)T+(55.8+55.8i)T2 1 + (13.0 + 5.38i)T + (55.8 + 55.8i)T^{2}
83 1+(−9.66−9.66i)T+83iT2 1 + (-9.66 - 9.66i)T + 83iT^{2}
89 1−15.1iT−89T2 1 - 15.1iT - 89T^{2}
97 1+(3.88+9.38i)T+(−68.5+68.5i)T2 1 + (3.88 + 9.38i)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

−10.91589872585281145395638828853, −10.06936082742225571433202756486, −9.498067057291345847051429706579, −8.198359074314996428352868358481, −6.58946005285369288172120259089, −5.96043833019327044272499155189, −5.35180946589548346677194519154, −4.51601485314832559761020369348, −2.64049591438848798792472686972, −0.29548133722556982024477923190, 1.73663167123277917331861910310, 3.77396365253067567946449859665, 4.36742025342853272684269077944, 5.91068977145071595094012526823, 6.76096620236370706363451016166, 7.44036672998628270670515243398, 8.518675066996283767933019198382, 10.09671394016019484474576609775, 10.65562035571181560415556043395, 11.77129534373019466580256268126

Graph of the ZZ-function along the critical line