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Properties

Label 2-325-5.4-c5-0-23
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  − 9.60i·2-s − 25.9i·3-s − 60.1·4-s − 248.·6-s + 181. i·7-s + 270. i·8-s − 428.·9-s + 512.·11-s + 1.56e3i·12-s − 169i·13-s + 1.74e3·14-s + 672.·16-s + 1.66e3i·17-s + 4.11e3i·18-s + 464.·19-s + ⋯
L(s)  = 1  − 1.69i·2-s − 1.66i·3-s − 1.88·4-s − 2.82·6-s + 1.40i·7-s + 1.49i·8-s − 1.76·9-s + 1.27·11-s + 3.12i·12-s − 0.277i·13-s + 2.37·14-s + 0.656·16-s + 1.39i·17-s + 2.99i·18-s + 0.294·19-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 1.5598174331.559817433
L(12)L(\frac12) ≈\approx 1.5598174331.559817433
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+9.60iT−32T2 1 + 9.60iT - 32T^{2}
3 1+25.9iT−243T2 1 + 25.9iT - 243T^{2}
7 1−181.iT−1.68e4T2 1 - 181. iT - 1.68e4T^{2}
11 1−512.T+1.61e5T2 1 - 512.T + 1.61e5T^{2}
17 1−1.66e3iT−1.41e6T2 1 - 1.66e3iT - 1.41e6T^{2}
19 1−464.T+2.47e6T2 1 - 464.T + 2.47e6T^{2}
23 1−2.09e3iT−6.43e6T2 1 - 2.09e3iT - 6.43e6T^{2}
29 1+5.93e3T+2.05e7T2 1 + 5.93e3T + 2.05e7T^{2}
31 1−8.39e3T+2.86e7T2 1 - 8.39e3T + 2.86e7T^{2}
37 1+8.90e3iT−6.93e7T2 1 + 8.90e3iT - 6.93e7T^{2}
41 1−3.25e3T+1.15e8T2 1 - 3.25e3T + 1.15e8T^{2}
43 1−5.95e3iT−1.47e8T2 1 - 5.95e3iT - 1.47e8T^{2}
47 1+1.07e4iT−2.29e8T2 1 + 1.07e4iT - 2.29e8T^{2}
53 1−3.93e4iT−4.18e8T2 1 - 3.93e4iT - 4.18e8T^{2}
59 1−3.94e3T+7.14e8T2 1 - 3.94e3T + 7.14e8T^{2}
61 1+4.14e4T+8.44e8T2 1 + 4.14e4T + 8.44e8T^{2}
67 1−1.85e4iT−1.35e9T2 1 - 1.85e4iT - 1.35e9T^{2}
71 1−7.03e4T+1.80e9T2 1 - 7.03e4T + 1.80e9T^{2}
73 1−7.86e4iT−2.07e9T2 1 - 7.86e4iT - 2.07e9T^{2}
79 1−7.17e4T+3.07e9T2 1 - 7.17e4T + 3.07e9T^{2}
83 1+208.iT−3.93e9T2 1 + 208. iT - 3.93e9T^{2}
89 1−4.35e4T+5.58e9T2 1 - 4.35e4T + 5.58e9T^{2}
97 1−1.52e4iT−8.58e9T2 1 - 1.52e4iT - 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

−10.89146354922169951510757112346, −9.476142008553620246253101588813, −8.820113150144340549271879017196, −7.87364698156272260791929868705, −6.45924533345049344114644287252, −5.59928607361801479129843137393, −3.83967728628371824602506533693, −2.63008633393877599448307514883, −1.79114366627448945750490698218, −1.03658712336380190183131111077, 0.49986923259793733657225560343, 3.51620124747826154007512723749, 4.42463094303585851322394886075, 4.94456368766598374432465716330, 6.30931917808127686327597705953, 7.09915689264292416481870545801, 8.193440158638402609765536147099, 9.307825465624314241780469731357, 9.676248607722121466522662707605, 10.80076137437969879670734338116

Graph of the ZZ-function along the critical line