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Properties

Label 2-425-17.13-c1-0-9
Degree 22
Conductor 425425
Sign −0.323−0.946i-0.323 - 0.946i
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  + 1.44i·2-s + (1.14 + 1.14i)3-s − 0.0943·4-s + (−1.65 + 1.65i)6-s + (2.69 − 2.69i)7-s + 2.75i·8-s − 0.370i·9-s + (−4.34 + 4.34i)11-s + (−0.108 − 0.108i)12-s + 2.56·13-s + (3.90 + 3.90i)14-s − 4.17·16-s + (3.45 + 2.25i)17-s + 0.535·18-s − 1.17i·19-s + ⋯
L(s)  = 1  + 1.02i·2-s + (0.662 + 0.662i)3-s − 0.0471·4-s + (−0.677 + 0.677i)6-s + (1.01 − 1.01i)7-s + 0.975i·8-s − 0.123i·9-s + (−1.31 + 1.31i)11-s + (−0.0312 − 0.0312i)12-s + 0.712·13-s + (1.04 + 1.04i)14-s − 1.04·16-s + (0.837 + 0.546i)17-s + 0.126·18-s − 0.268i·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.323−0.946i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.323−0.946i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.323−0.946i-0.323 - 0.946i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(251,⋅)\chi_{425} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), −0.323−0.946i)(2,\ 425,\ (\ :1/2),\ -0.323 - 0.946i)

Particular Values

L(1)L(1) ≈\approx 1.15276+1.61207i1.15276 + 1.61207i
L(12)L(\frac12) ≈\approx 1.15276+1.61207i1.15276 + 1.61207i
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+(−3.45−2.25i)T 1 + (-3.45 - 2.25i)T
good2 1−1.44iT−2T2 1 - 1.44iT - 2T^{2}
3 1+(−1.14−1.14i)T+3iT2 1 + (-1.14 - 1.14i)T + 3iT^{2}
7 1+(−2.69+2.69i)T−7iT2 1 + (-2.69 + 2.69i)T - 7iT^{2}
11 1+(4.34−4.34i)T−11iT2 1 + (4.34 - 4.34i)T - 11iT^{2}
13 1−2.56T+13T2 1 - 2.56T + 13T^{2}
19 1+1.17iT−19T2 1 + 1.17iT - 19T^{2}
23 1+(2.53−2.53i)T−23iT2 1 + (2.53 - 2.53i)T - 23iT^{2}
29 1+(3.70+3.70i)T+29iT2 1 + (3.70 + 3.70i)T + 29iT^{2}
31 1+(0.394+0.394i)T+31iT2 1 + (0.394 + 0.394i)T + 31iT^{2}
37 1+(5.28+5.28i)T+37iT2 1 + (5.28 + 5.28i)T + 37iT^{2}
41 1+(−5.35+5.35i)T−41iT2 1 + (-5.35 + 5.35i)T - 41iT^{2}
43 1+0.774iT−43T2 1 + 0.774iT - 43T^{2}
47 1+4.35T+47T2 1 + 4.35T + 47T^{2}
53 1+2.36iT−53T2 1 + 2.36iT - 53T^{2}
59 1+12.8iT−59T2 1 + 12.8iT - 59T^{2}
61 1+(3.29−3.29i)T−61iT2 1 + (3.29 - 3.29i)T - 61iT^{2}
67 1−2.97T+67T2 1 - 2.97T + 67T^{2}
71 1+(5.97+5.97i)T+71iT2 1 + (5.97 + 5.97i)T + 71iT^{2}
73 1+(−11.8−11.8i)T+73iT2 1 + (-11.8 - 11.8i)T + 73iT^{2}
79 1+(8.09−8.09i)T−79iT2 1 + (8.09 - 8.09i)T - 79iT^{2}
83 1+5.07iT−83T2 1 + 5.07iT - 83T^{2}
89 1−8.16T+89T2 1 - 8.16T + 89T^{2}
97 1+(−5.16−5.16i)T+97iT2 1 + (-5.16 - 5.16i)T + 97iT^{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

−11.17420242209977652226438458364, −10.46042471812034948817667960545, −9.599908661214685766340455896180, −8.341149325795085322106939068641, −7.78019026413272347754176913525, −7.08443874357816554384946871464, −5.70795990995273183425861186695, −4.74142250743265467706583059439, −3.73090942394239693437946802141, −2.05266271660104069332269535591, 1.43050939773521367587640350199, 2.52679844393801244346181426905, 3.27571125808881206486213684521, 5.06245547181340853119024272450, 6.05876625857836852627506870182, 7.56352408064781206025976111318, 8.228802298829311473723498467548, 8.951358682220416187696009733566, 10.32995437726502018856885192454, 11.01302443433801000365185691218

Graph of the ZZ-function along the critical line