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Properties

Label 2-325-5.4-c5-0-57
Degree 22
Conductor 325325
Sign 0.894−0.447i0.894 - 0.447i
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  + 2.16i·2-s − 2.56i·3-s + 27.2·4-s + 5.57·6-s + 75.5i·7-s + 128. i·8-s + 236.·9-s + 624.·11-s − 70.1i·12-s − 169i·13-s − 163.·14-s + 594.·16-s − 2.34e3i·17-s + 512. i·18-s + 283.·19-s + ⋯
L(s)  = 1  + 0.383i·2-s − 0.164i·3-s + 0.853·4-s + 0.0631·6-s + 0.583i·7-s + 0.710i·8-s + 0.972·9-s + 1.55·11-s − 0.140i·12-s − 0.277i·13-s − 0.223·14-s + 0.580·16-s − 1.96i·17-s + 0.372i·18-s + 0.180·19-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+5/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: 0.894−0.447i0.894 - 0.447i
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), 0.894−0.447i)(2,\ 325,\ (\ :5/2),\ 0.894 - 0.447i)

Particular Values

L(3)L(3) ≈\approx 3.3342479283.334247928
L(12)L(\frac12) ≈\approx 3.3342479283.334247928
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 1−2.16iT−32T2 1 - 2.16iT - 32T^{2}
3 1+2.56iT−243T2 1 + 2.56iT - 243T^{2}
7 1−75.5iT−1.68e4T2 1 - 75.5iT - 1.68e4T^{2}
11 1−624.T+1.61e5T2 1 - 624.T + 1.61e5T^{2}
17 1+2.34e3iT−1.41e6T2 1 + 2.34e3iT - 1.41e6T^{2}
19 1−283.T+2.47e6T2 1 - 283.T + 2.47e6T^{2}
23 1−2.04e3iT−6.43e6T2 1 - 2.04e3iT - 6.43e6T^{2}
29 1+6.17e3T+2.05e7T2 1 + 6.17e3T + 2.05e7T^{2}
31 1−687.T+2.86e7T2 1 - 687.T + 2.86e7T^{2}
37 1−2.79e3iT−6.93e7T2 1 - 2.79e3iT - 6.93e7T^{2}
41 1−8.23e3T+1.15e8T2 1 - 8.23e3T + 1.15e8T^{2}
43 1+1.32e4iT−1.47e8T2 1 + 1.32e4iT - 1.47e8T^{2}
47 1−1.54e4iT−2.29e8T2 1 - 1.54e4iT - 2.29e8T^{2}
53 1+9.60e3iT−4.18e8T2 1 + 9.60e3iT - 4.18e8T^{2}
59 1+4.01e4T+7.14e8T2 1 + 4.01e4T + 7.14e8T^{2}
61 1−3.25e4T+8.44e8T2 1 - 3.25e4T + 8.44e8T^{2}
67 1+1.59e4iT−1.35e9T2 1 + 1.59e4iT - 1.35e9T^{2}
71 1−6.02e4T+1.80e9T2 1 - 6.02e4T + 1.80e9T^{2}
73 1+3.54e4iT−2.07e9T2 1 + 3.54e4iT - 2.07e9T^{2}
79 1+6.50e4T+3.07e9T2 1 + 6.50e4T + 3.07e9T^{2}
83 1−8.60e4iT−3.93e9T2 1 - 8.60e4iT - 3.93e9T^{2}
89 1−1.39e5T+5.58e9T2 1 - 1.39e5T + 5.58e9T^{2}
97 1−8.01e3iT−8.58e9T2 1 - 8.01e3iT - 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

−11.08031277700771620970481198719, −9.707209800225198220755367452589, −9.081124816873889724971641964821, −7.62927986632082208265675385195, −7.06413692121648539715785665848, −6.13354678865165297277288440490, −5.05272226834863098845138913528, −3.59005226691299749399250376579, −2.24389808082366553954672723115, −1.09635142873937471557404468002, 1.11956508238631164664216364784, 1.91194990697267016358651702744, 3.68698116314434204709597803059, 4.19219122233422467571212945588, 6.07637490598718414916003825019, 6.78555559797945081866672712560, 7.68367465747946515256152065341, 9.019578055159362458442436782815, 10.01496731917669498611712319077, 10.70405250720178784298019422753

Graph of the ZZ-function along the critical line