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Properties

Label 2-325-5.4-c5-0-30
Degree 22
Conductor 325325
Sign 0.447−0.894i0.447 - 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  + 1.83i·2-s − 11.9i·3-s + 28.6·4-s + 21.9·6-s + 112. i·7-s + 111. i·8-s + 100.·9-s + 162.·11-s − 341. i·12-s + 169i·13-s − 206.·14-s + 711.·16-s + 379. i·17-s + 185. i·18-s + 284.·19-s + ⋯
L(s)  = 1  + 0.324i·2-s − 0.765i·3-s + 0.894·4-s + 0.248·6-s + 0.865i·7-s + 0.615i·8-s + 0.414·9-s + 0.405·11-s − 0.684i·12-s + 0.277i·13-s − 0.281·14-s + 0.694·16-s + 0.318i·17-s + 0.134i·18-s + 0.180·19-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: 0.447−0.894i0.447 - 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 2.6953899692.695389969
L(12)L(\frac12) ≈\approx 2.6953899692.695389969
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 1−1.83iT−32T2 1 - 1.83iT - 32T^{2}
3 1+11.9iT−243T2 1 + 11.9iT - 243T^{2}
7 1−112.iT−1.68e4T2 1 - 112. iT - 1.68e4T^{2}
11 1−162.T+1.61e5T2 1 - 162.T + 1.61e5T^{2}
17 1−379.iT−1.41e6T2 1 - 379. iT - 1.41e6T^{2}
19 1−284.T+2.47e6T2 1 - 284.T + 2.47e6T^{2}
23 1−4.18e3iT−6.43e6T2 1 - 4.18e3iT - 6.43e6T^{2}
29 1+6.27e3T+2.05e7T2 1 + 6.27e3T + 2.05e7T^{2}
31 1+7.55e3T+2.86e7T2 1 + 7.55e3T + 2.86e7T^{2}
37 1−7.06e3iT−6.93e7T2 1 - 7.06e3iT - 6.93e7T^{2}
41 1−4.16e3T+1.15e8T2 1 - 4.16e3T + 1.15e8T^{2}
43 1−2.15e4iT−1.47e8T2 1 - 2.15e4iT - 1.47e8T^{2}
47 1+2.26e4iT−2.29e8T2 1 + 2.26e4iT - 2.29e8T^{2}
53 1−4.77e3iT−4.18e8T2 1 - 4.77e3iT - 4.18e8T^{2}
59 1−4.83e4T+7.14e8T2 1 - 4.83e4T + 7.14e8T^{2}
61 1−3.65e3T+8.44e8T2 1 - 3.65e3T + 8.44e8T^{2}
67 1−1.58e4iT−1.35e9T2 1 - 1.58e4iT - 1.35e9T^{2}
71 1+3.72e4T+1.80e9T2 1 + 3.72e4T + 1.80e9T^{2}
73 1+2.93e4iT−2.07e9T2 1 + 2.93e4iT - 2.07e9T^{2}
79 1−4.49e4T+3.07e9T2 1 - 4.49e4T + 3.07e9T^{2}
83 1−2.37e4iT−3.93e9T2 1 - 2.37e4iT - 3.93e9T^{2}
89 1+561.T+5.58e9T2 1 + 561.T + 5.58e9T^{2}
97 1−3.12e4iT−8.58e9T2 1 - 3.12e4iT - 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

−11.30280323798401902851740368916, −9.951403606597783620156376720167, −8.943117663102984580528734133793, −7.76688859930420890803411724263, −7.14808661888949397527802209113, −6.18864393037342419137371164722, −5.37621896241210634047300455149, −3.61534640569278987666313497730, −2.17912972510250573612173677230, −1.41641279620037327964612203024, 0.68573794385259305979094688765, 2.05023243237425102238506974366, 3.50124925730503693395997566675, 4.24468832922379269888108130631, 5.62021496414843989268539560769, 6.92560150943966331770121468514, 7.49427588879352611399086282803, 8.993475696879959283018405981942, 9.952694662998735051047687477136, 10.67024502496290920114450006736

Graph of the ZZ-function along the critical line