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Properties

Label 2-325-5.4-c5-0-54
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  + 8.11i·2-s − 3.22i·3-s − 33.8·4-s + 26.2·6-s + 20.7i·7-s − 15.2i·8-s + 232.·9-s + 134.·11-s + 109. i·12-s + 169i·13-s − 168.·14-s − 960.·16-s − 2.19e3i·17-s + 1.88e3i·18-s + 1.98e3·19-s + ⋯
L(s)  = 1  + 1.43i·2-s − 0.207i·3-s − 1.05·4-s + 0.297·6-s + 0.159i·7-s − 0.0844i·8-s + 0.957·9-s + 0.335·11-s + 0.219i·12-s + 0.277i·13-s − 0.229·14-s − 0.937·16-s − 1.84i·17-s + 1.37i·18-s + 1.25·19-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 2.2521419602.252141960
L(12)L(\frac12) ≈\approx 2.2521419602.252141960
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 1−8.11iT−32T2 1 - 8.11iT - 32T^{2}
3 1+3.22iT−243T2 1 + 3.22iT - 243T^{2}
7 1−20.7iT−1.68e4T2 1 - 20.7iT - 1.68e4T^{2}
11 1−134.T+1.61e5T2 1 - 134.T + 1.61e5T^{2}
17 1+2.19e3iT−1.41e6T2 1 + 2.19e3iT - 1.41e6T^{2}
19 1−1.98e3T+2.47e6T2 1 - 1.98e3T + 2.47e6T^{2}
23 1+4.22e3iT−6.43e6T2 1 + 4.22e3iT - 6.43e6T^{2}
29 1+2.54e3T+2.05e7T2 1 + 2.54e3T + 2.05e7T^{2}
31 1+1.36e3T+2.86e7T2 1 + 1.36e3T + 2.86e7T^{2}
37 1+1.41e4iT−6.93e7T2 1 + 1.41e4iT - 6.93e7T^{2}
41 1−1.17e4T+1.15e8T2 1 - 1.17e4T + 1.15e8T^{2}
43 1−6.62e3iT−1.47e8T2 1 - 6.62e3iT - 1.47e8T^{2}
47 1−1.48e4iT−2.29e8T2 1 - 1.48e4iT - 2.29e8T^{2}
53 1−2.42e4iT−4.18e8T2 1 - 2.42e4iT - 4.18e8T^{2}
59 1+1.43e3T+7.14e8T2 1 + 1.43e3T + 7.14e8T^{2}
61 1−1.62e4T+8.44e8T2 1 - 1.62e4T + 8.44e8T^{2}
67 1+1.67e4iT−1.35e9T2 1 + 1.67e4iT - 1.35e9T^{2}
71 1−2.71e4T+1.80e9T2 1 - 2.71e4T + 1.80e9T^{2}
73 1−6.32e4iT−2.07e9T2 1 - 6.32e4iT - 2.07e9T^{2}
79 1−5.81e4T+3.07e9T2 1 - 5.81e4T + 3.07e9T^{2}
83 1+1.21e5iT−3.93e9T2 1 + 1.21e5iT - 3.93e9T^{2}
89 1−4.98e4T+5.58e9T2 1 - 4.98e4T + 5.58e9T^{2}
97 1−2.13e4iT−8.58e9T2 1 - 2.13e4iT - 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

−10.95502841591742620273268747690, −9.545075841507111889019881929559, −8.994183458009201841098726822970, −7.60479194631040237126053437342, −7.23328775008620077425614016519, −6.26043740264983442479912369523, −5.19288597930751357174712064821, −4.26138279276581883662092776084, −2.47038931253312622074028517608, −0.72583150274931120706694165845, 1.05602310791307346173127558315, 1.85393009214672584364261957931, 3.44799612437523565174066540821, 3.99137673253118656230178202746, 5.35931966135605886508121578247, 6.78783614182216681316962834359, 7.897020832011317533456648726660, 9.235300872652246705587873852888, 9.923672135946090102210867496329, 10.60109323713698264517285800830

Graph of the ZZ-function along the critical line