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Properties

Label 2-325-5.4-c5-0-75
Degree 22
Conductor 325325
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 52.124752.1247
Root an. cond. 7.219747.21974
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.979i·2-s − 20.7i·3-s + 31.0·4-s + 20.3·6-s − 233. i·7-s + 61.7i·8-s − 188.·9-s + 621.·11-s − 645. i·12-s − 169i·13-s + 229.·14-s + 932.·16-s + 287. i·17-s − 185. i·18-s + 2.63e3·19-s + ⋯
L(s)  = 1  + 0.173i·2-s − 1.33i·3-s + 0.970·4-s + 0.230·6-s − 1.80i·7-s + 0.341i·8-s − 0.777·9-s + 1.54·11-s − 1.29i·12-s − 0.277i·13-s + 0.312·14-s + 0.910·16-s + 0.241i·17-s − 0.134i·18-s + 1.67·19-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+5/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 52.124752.1247
Root analytic conductor: 7.219747.21974
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ325(274,⋅)\chi_{325} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :5/2), −0.447+0.894i)(2,\ 325,\ (\ :5/2),\ -0.447 + 0.894i)

Particular Values

L(3)L(3) ≈\approx 3.2967095053.296709505
L(12)L(\frac12) ≈\approx 3.2967095053.296709505
L(72)L(\frac{7}{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
13 1+169iT 1 + 169iT
good2 1−0.979iT−32T2 1 - 0.979iT - 32T^{2}
3 1+20.7iT−243T2 1 + 20.7iT - 243T^{2}
7 1+233.iT−1.68e4T2 1 + 233. iT - 1.68e4T^{2}
11 1−621.T+1.61e5T2 1 - 621.T + 1.61e5T^{2}
17 1−287.iT−1.41e6T2 1 - 287. iT - 1.41e6T^{2}
19 1−2.63e3T+2.47e6T2 1 - 2.63e3T + 2.47e6T^{2}
23 1+2.16e3iT−6.43e6T2 1 + 2.16e3iT - 6.43e6T^{2}
29 1+2.47e3T+2.05e7T2 1 + 2.47e3T + 2.05e7T^{2}
31 1−7.14e3T+2.86e7T2 1 - 7.14e3T + 2.86e7T^{2}
37 1−2.72e3iT−6.93e7T2 1 - 2.72e3iT - 6.93e7T^{2}
41 1−6.10e3T+1.15e8T2 1 - 6.10e3T + 1.15e8T^{2}
43 1−2.66e3iT−1.47e8T2 1 - 2.66e3iT - 1.47e8T^{2}
47 1−2.96e4iT−2.29e8T2 1 - 2.96e4iT - 2.29e8T^{2}
53 1−1.17e4iT−4.18e8T2 1 - 1.17e4iT - 4.18e8T^{2}
59 1+2.86e4T+7.14e8T2 1 + 2.86e4T + 7.14e8T^{2}
61 1−3.05e4T+8.44e8T2 1 - 3.05e4T + 8.44e8T^{2}
67 1−2.21e4iT−1.35e9T2 1 - 2.21e4iT - 1.35e9T^{2}
71 1+6.24e4T+1.80e9T2 1 + 6.24e4T + 1.80e9T^{2}
73 1+4.05e4iT−2.07e9T2 1 + 4.05e4iT - 2.07e9T^{2}
79 1−1.89e3T+3.07e9T2 1 - 1.89e3T + 3.07e9T^{2}
83 1−6.63e3iT−3.93e9T2 1 - 6.63e3iT - 3.93e9T^{2}
89 1+1.39e5T+5.58e9T2 1 + 1.39e5T + 5.58e9T^{2}
97 1−5.17e4iT−8.58e9T2 1 - 5.17e4iT - 8.58e9T^{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

−10.66291933283943845715751527035, −9.635434092614960902069930074413, −8.041822632914343321998881232948, −7.40166582199878225720261890233, −6.77658661788774618546610641965, −6.09369214696144003248485699046, −4.26386671876544906782270772070, −2.98492849191074524342747583336, −1.33855443233139415546544713138, −0.995725041551171598062425769368, 1.55174508312849642334074035703, 2.86838968975088413434905600381, 3.78133814753018525322969028379, 5.20749665453848066017032494680, 5.97502435262321941565315612826, 7.12768673117038360233446024931, 8.616230365670838013039490768157, 9.452113743394712015930586541737, 9.907777083612065547924707188597, 11.42933116723221701689425382394

Graph of the ZZ-function along the critical line