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Properties

Label 2-325-5.4-c5-0-72
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  + 8.23i·2-s − 11.6i·3-s − 35.8·4-s + 95.8·6-s − 195. i·7-s − 31.9i·8-s + 107.·9-s + 64.5·11-s + 417. i·12-s − 169i·13-s + 1.60e3·14-s − 885.·16-s + 426. i·17-s + 886. i·18-s + 959.·19-s + ⋯
L(s)  = 1  + 1.45i·2-s − 0.746i·3-s − 1.12·4-s + 1.08·6-s − 1.50i·7-s − 0.176i·8-s + 0.442·9-s + 0.160·11-s + 0.836i·12-s − 0.277i·13-s + 2.19·14-s − 0.864·16-s + 0.357i·17-s + 0.644i·18-s + 0.609·19-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 1.2880719401.288071940
L(12)L(\frac12) ≈\approx 1.2880719401.288071940
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 1−8.23iT−32T2 1 - 8.23iT - 32T^{2}
3 1+11.6iT−243T2 1 + 11.6iT - 243T^{2}
7 1+195.iT−1.68e4T2 1 + 195. iT - 1.68e4T^{2}
11 1−64.5T+1.61e5T2 1 - 64.5T + 1.61e5T^{2}
17 1−426.iT−1.41e6T2 1 - 426. iT - 1.41e6T^{2}
19 1−959.T+2.47e6T2 1 - 959.T + 2.47e6T^{2}
23 1−499.iT−6.43e6T2 1 - 499. iT - 6.43e6T^{2}
29 1+1.28e3T+2.05e7T2 1 + 1.28e3T + 2.05e7T^{2}
31 1+6.73e3T+2.86e7T2 1 + 6.73e3T + 2.86e7T^{2}
37 1+6.21e3iT−6.93e7T2 1 + 6.21e3iT - 6.93e7T^{2}
41 1+6.49e3T+1.15e8T2 1 + 6.49e3T + 1.15e8T^{2}
43 1−1.56e4iT−1.47e8T2 1 - 1.56e4iT - 1.47e8T^{2}
47 1+6.29e3iT−2.29e8T2 1 + 6.29e3iT - 2.29e8T^{2}
53 1+4.03e4iT−4.18e8T2 1 + 4.03e4iT - 4.18e8T^{2}
59 1+2.56e4T+7.14e8T2 1 + 2.56e4T + 7.14e8T^{2}
61 1−2.41e4T+8.44e8T2 1 - 2.41e4T + 8.44e8T^{2}
67 1+3.91e4iT−1.35e9T2 1 + 3.91e4iT - 1.35e9T^{2}
71 1+3.26e4T+1.80e9T2 1 + 3.26e4T + 1.80e9T^{2}
73 1+1.45e4iT−2.07e9T2 1 + 1.45e4iT - 2.07e9T^{2}
79 1+7.90e4T+3.07e9T2 1 + 7.90e4T + 3.07e9T^{2}
83 1+1.02e5iT−3.93e9T2 1 + 1.02e5iT - 3.93e9T^{2}
89 1−4.81e4T+5.58e9T2 1 - 4.81e4T + 5.58e9T^{2}
97 1−7.33e4iT−8.58e9T2 1 - 7.33e4iT - 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

−10.50664813505657075611510300830, −9.467936666385442479415893418167, −8.180687562170295818571829472961, −7.43468974872750279001660989422, −7.00195435856139709439648558374, −6.05107153719232471121279053808, −4.83491302887465145350715053809, −3.70807636165936620376841431572, −1.64530475903753231304915022638, −0.33727142883327324018425899725, 1.43612129908459935773676515898, 2.53193075464804179745251566112, 3.54070962695427092005116606982, 4.62198387361713740041410107573, 5.65159343388496543369243151133, 7.12321734301169355654095133457, 8.775002564156947598188992787146, 9.324518292813121001400524569104, 10.06638045767113916921674940028, 10.99224801499722497449852765273

Graph of the ZZ-function along the critical line