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Properties

Label 2-325-5.4-c5-0-71
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  + 6.07i·2-s − 25.6i·3-s − 4.94·4-s + 155.·6-s + 137. i·7-s + 164. i·8-s − 413.·9-s + 169.·11-s + 126. i·12-s − 169i·13-s − 834.·14-s − 1.15e3·16-s − 487. i·17-s − 2.51e3i·18-s − 2.38e3·19-s + ⋯
L(s)  = 1  + 1.07i·2-s − 1.64i·3-s − 0.154·4-s + 1.76·6-s + 1.05i·7-s + 0.908i·8-s − 1.70·9-s + 0.421·11-s + 0.254i·12-s − 0.277i·13-s − 1.13·14-s − 1.13·16-s − 0.408i·17-s − 1.82i·18-s − 1.51·19-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 0.83431811780.8343181178
L(12)L(\frac12) ≈\approx 0.83431811780.8343181178
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 1−6.07iT−32T2 1 - 6.07iT - 32T^{2}
3 1+25.6iT−243T2 1 + 25.6iT - 243T^{2}
7 1−137.iT−1.68e4T2 1 - 137. iT - 1.68e4T^{2}
11 1−169.T+1.61e5T2 1 - 169.T + 1.61e5T^{2}
17 1+487.iT−1.41e6T2 1 + 487. iT - 1.41e6T^{2}
19 1+2.38e3T+2.47e6T2 1 + 2.38e3T + 2.47e6T^{2}
23 1+4.14e3iT−6.43e6T2 1 + 4.14e3iT - 6.43e6T^{2}
29 1−4.57e3T+2.05e7T2 1 - 4.57e3T + 2.05e7T^{2}
31 1+2.95e3T+2.86e7T2 1 + 2.95e3T + 2.86e7T^{2}
37 1−5.73e3iT−6.93e7T2 1 - 5.73e3iT - 6.93e7T^{2}
41 1+1.70e3T+1.15e8T2 1 + 1.70e3T + 1.15e8T^{2}
43 1+1.09e4iT−1.47e8T2 1 + 1.09e4iT - 1.47e8T^{2}
47 1+7.91e3iT−2.29e8T2 1 + 7.91e3iT - 2.29e8T^{2}
53 1+2.64e4iT−4.18e8T2 1 + 2.64e4iT - 4.18e8T^{2}
59 1+1.12e4T+7.14e8T2 1 + 1.12e4T + 7.14e8T^{2}
61 1+4.02e4T+8.44e8T2 1 + 4.02e4T + 8.44e8T^{2}
67 1+6.48e4iT−1.35e9T2 1 + 6.48e4iT - 1.35e9T^{2}
71 1+4.70e4T+1.80e9T2 1 + 4.70e4T + 1.80e9T^{2}
73 1+4.27e4iT−2.07e9T2 1 + 4.27e4iT - 2.07e9T^{2}
79 1+9.00e4T+3.07e9T2 1 + 9.00e4T + 3.07e9T^{2}
83 1−1.64e4iT−3.93e9T2 1 - 1.64e4iT - 3.93e9T^{2}
89 1−3.56e4T+5.58e9T2 1 - 3.56e4T + 5.58e9T^{2}
97 1+1.21e5iT−8.58e9T2 1 + 1.21e5iT - 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

−10.63629028761202070187271341357, −8.776387954484754840817112145412, −8.439210623721892381742409479095, −7.42874489963799323343533642504, −6.45108318449258411226025877962, −6.18170812831406850587190191681, −4.91778359903567618563532810606, −2.66653389151328845499909775463, −1.87515480593145921068027818920, −0.20320156896442999409901519302, 1.41823243721137462808196812323, 2.99144136971711744263983233692, 4.01848971009343847533116733737, 4.38928529926288370143696484975, 6.00034906093732428536283742014, 7.26961981637941221000800025739, 8.777312801857995170595357600103, 9.621421253475044928351954759514, 10.33886649888557775465129395205, 10.87874385118669052686812728114

Graph of the ZZ-function along the critical line