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Properties

Label 2-425-85.19-c1-0-7
Degree 22
Conductor 425425
Sign −0.637−0.770i-0.637 - 0.770i
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  + (−1.66 + 1.66i)2-s + (0.600 + 1.44i)3-s − 3.53i·4-s + (−3.41 − 1.41i)6-s + (3.30 + 1.37i)7-s + (2.54 + 2.54i)8-s + (0.379 − 0.379i)9-s + (2.29 + 0.950i)11-s + (5.12 − 2.12i)12-s + 1.25·13-s + (−7.78 + 3.22i)14-s − 1.40·16-s + (1.48 − 3.84i)17-s + 1.26i·18-s + (2.56 + 2.56i)19-s + ⋯
L(s)  = 1  + (−1.17 + 1.17i)2-s + (0.346 + 0.837i)3-s − 1.76i·4-s + (−1.39 − 0.576i)6-s + (1.25 + 0.518i)7-s + (0.900 + 0.900i)8-s + (0.126 − 0.126i)9-s + (0.691 + 0.286i)11-s + (1.47 − 0.612i)12-s + 0.349·13-s + (−2.08 + 0.861i)14-s − 0.352·16-s + (0.360 − 0.932i)17-s + 0.297i·18-s + (0.589 + 0.589i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.637−0.770i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.637−0.770i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.637−0.770i-0.637 - 0.770i
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), −0.637−0.770i)(2,\ 425,\ (\ :1/2),\ -0.637 - 0.770i)

Particular Values

L(1)L(1) ≈\approx 0.441209+0.938001i0.441209 + 0.938001i
L(12)L(\frac12) ≈\approx 0.441209+0.938001i0.441209 + 0.938001i
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+(−1.48+3.84i)T 1 + (-1.48 + 3.84i)T
good2 1+(1.66−1.66i)T−2iT2 1 + (1.66 - 1.66i)T - 2iT^{2}
3 1+(−0.600−1.44i)T+(−2.12+2.12i)T2 1 + (-0.600 - 1.44i)T + (-2.12 + 2.12i)T^{2}
7 1+(−3.30−1.37i)T+(4.94+4.94i)T2 1 + (-3.30 - 1.37i)T + (4.94 + 4.94i)T^{2}
11 1+(−2.29−0.950i)T+(7.77+7.77i)T2 1 + (-2.29 - 0.950i)T + (7.77 + 7.77i)T^{2}
13 1−1.25T+13T2 1 - 1.25T + 13T^{2}
19 1+(−2.56−2.56i)T+19iT2 1 + (-2.56 - 2.56i)T + 19iT^{2}
23 1+(3.38−8.16i)T+(−16.2−16.2i)T2 1 + (3.38 - 8.16i)T + (-16.2 - 16.2i)T^{2}
29 1+(3.35+8.09i)T+(−20.5+20.5i)T2 1 + (3.35 + 8.09i)T + (-20.5 + 20.5i)T^{2}
31 1+(−2.25+0.935i)T+(21.9−21.9i)T2 1 + (-2.25 + 0.935i)T + (21.9 - 21.9i)T^{2}
37 1+(1.76+4.25i)T+(−26.1+26.1i)T2 1 + (1.76 + 4.25i)T + (-26.1 + 26.1i)T^{2}
41 1+(1.65−3.99i)T+(−28.9−28.9i)T2 1 + (1.65 - 3.99i)T + (-28.9 - 28.9i)T^{2}
43 1+(5.33+5.33i)T+43iT2 1 + (5.33 + 5.33i)T + 43iT^{2}
47 1+11.3T+47T2 1 + 11.3T + 47T^{2}
53 1+(−4.02+4.02i)T−53iT2 1 + (-4.02 + 4.02i)T - 53iT^{2}
59 1+(3.16−3.16i)T−59iT2 1 + (3.16 - 3.16i)T - 59iT^{2}
61 1+(−0.0929+0.224i)T+(−43.1−43.1i)T2 1 + (-0.0929 + 0.224i)T + (-43.1 - 43.1i)T^{2}
67 1−7.23iT−67T2 1 - 7.23iT - 67T^{2}
71 1+(1.69−0.703i)T+(50.2−50.2i)T2 1 + (1.69 - 0.703i)T + (50.2 - 50.2i)T^{2}
73 1+(5.05−2.09i)T+(51.6−51.6i)T2 1 + (5.05 - 2.09i)T + (51.6 - 51.6i)T^{2}
79 1+(−8.34−3.45i)T+(55.8+55.8i)T2 1 + (-8.34 - 3.45i)T + (55.8 + 55.8i)T^{2}
83 1+(5.17−5.17i)T−83iT2 1 + (5.17 - 5.17i)T - 83iT^{2}
89 1−2.19iT−89T2 1 - 2.19iT - 89T^{2}
97 1+(−9.07+3.75i)T+(68.5−68.5i)T2 1 + (-9.07 + 3.75i)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

−11.35772640993556340274598720216, −9.891507138929912623505383501345, −9.680330906661907119777561878744, −8.723473117662685832882766442955, −7.970879003766134251272685788736, −7.18072066098933694449730091565, −5.91307984682464701119957823699, −5.03618059646237345819776431720, −3.72093389348558841984428772072, −1.53376318802040265692202563628, 1.18487065680015306737331041911, 1.88584523006573766673731459019, 3.34073138197306968055319683547, 4.70928135031962541908904857361, 6.54096067588431227104592406668, 7.64395174551498752230529619525, 8.290902486929160580536920618983, 8.871714434246994702568851406962, 10.19113486591191993502616134393, 10.77015528606560385727378166195

Graph of the ZZ-function along the critical line