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Properties

Label 2-325-5.4-c5-0-83
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  − 7.28i·2-s − 14.9i·3-s − 21.0·4-s − 109.·6-s − 135. i·7-s − 79.9i·8-s + 18.1·9-s + 191.·11-s + 315. i·12-s + 169i·13-s − 985.·14-s − 1.25e3·16-s + 874. i·17-s − 132. i·18-s − 1.99e3·19-s + ⋯
L(s)  = 1  − 1.28i·2-s − 0.961i·3-s − 0.656·4-s − 1.23·6-s − 1.04i·7-s − 0.441i·8-s + 0.0748·9-s + 0.476·11-s + 0.631i·12-s + 0.277i·13-s − 1.34·14-s − 1.22·16-s + 0.733i·17-s − 0.0963i·18-s − 1.26·19-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 1.1158339711.115833971
L(12)L(\frac12) ≈\approx 1.1158339711.115833971
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+7.28iT−32T2 1 + 7.28iT - 32T^{2}
3 1+14.9iT−243T2 1 + 14.9iT - 243T^{2}
7 1+135.iT−1.68e4T2 1 + 135. iT - 1.68e4T^{2}
11 1−191.T+1.61e5T2 1 - 191.T + 1.61e5T^{2}
17 1−874.iT−1.41e6T2 1 - 874. iT - 1.41e6T^{2}
19 1+1.99e3T+2.47e6T2 1 + 1.99e3T + 2.47e6T^{2}
23 1+2.09e3iT−6.43e6T2 1 + 2.09e3iT - 6.43e6T^{2}
29 1−46.3T+2.05e7T2 1 - 46.3T + 2.05e7T^{2}
31 1+9.45e3T+2.86e7T2 1 + 9.45e3T + 2.86e7T^{2}
37 1−3.37e3iT−6.93e7T2 1 - 3.37e3iT - 6.93e7T^{2}
41 1+1.05e4T+1.15e8T2 1 + 1.05e4T + 1.15e8T^{2}
43 1−4.05e3iT−1.47e8T2 1 - 4.05e3iT - 1.47e8T^{2}
47 1−1.70e4iT−2.29e8T2 1 - 1.70e4iT - 2.29e8T^{2}
53 1+2.39e4iT−4.18e8T2 1 + 2.39e4iT - 4.18e8T^{2}
59 1−1.01e4T+7.14e8T2 1 - 1.01e4T + 7.14e8T^{2}
61 1+2.74e4T+8.44e8T2 1 + 2.74e4T + 8.44e8T^{2}
67 1−317.iT−1.35e9T2 1 - 317. iT - 1.35e9T^{2}
71 1−3.96e4T+1.80e9T2 1 - 3.96e4T + 1.80e9T^{2}
73 1−4.76e3iT−2.07e9T2 1 - 4.76e3iT - 2.07e9T^{2}
79 1−7.45e4T+3.07e9T2 1 - 7.45e4T + 3.07e9T^{2}
83 1+1.42e4iT−3.93e9T2 1 + 1.42e4iT - 3.93e9T^{2}
89 1−1.32e5T+5.58e9T2 1 - 1.32e5T + 5.58e9T^{2}
97 1+1.93e4iT−8.58e9T2 1 + 1.93e4iT - 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

−10.39187597998844865574326374157, −9.308899570519870320329400091452, −8.117229863790768616422167241696, −6.98024535387071298113436536491, −6.41623335791145311470242217024, −4.42764840796111830302177476609, −3.63776908646797401063906132089, −2.11760821420929738348589662071, −1.37378004706762973458519324724, −0.28018021364497598991012094497, 2.11426362595565290552993169300, 3.70176720004124815817578196677, 4.94839816640159478449104223288, 5.62020375057735067447250347273, 6.67536926932661694379475342111, 7.65650558381964430119226894127, 8.873021483246194207374195342656, 9.232717286880752918463405580491, 10.50152883012026997341774828924, 11.42758280625037475533058061364

Graph of the ZZ-function along the critical line