Properties

Label 2-325-5.4-c5-0-79
Degree 22
Conductor 325325
Sign 0.894+0.447i-0.894 + 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  + 6.25i·2-s − 21.3i·3-s − 7.11·4-s + 133.·6-s − 116. i·7-s + 155. i·8-s − 213.·9-s − 259.·11-s + 152. i·12-s + 169i·13-s + 727.·14-s − 1.20e3·16-s − 876. i·17-s − 1.33e3i·18-s + 921.·19-s + ⋯
L(s)  = 1  + 1.10i·2-s − 1.37i·3-s − 0.222·4-s + 1.51·6-s − 0.897i·7-s + 0.859i·8-s − 0.880·9-s − 0.645·11-s + 0.305i·12-s + 0.277i·13-s + 0.992·14-s − 1.17·16-s − 0.735i·17-s − 0.973i·18-s + 0.585·19-s + ⋯

Functional equation

Λ(s)=(325s/2ΓC(s)L(s)=((0.894+0.447i)Λ(6s)\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/2ΓC(s+5/2)L(s)=((0.894+0.447i)Λ(1s)\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    =    52135^{2} \cdot 13
Sign: 0.894+0.447i-0.894 + 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 0.54529585530.5452958553
L(12)L(\frac12) \approx 0.54529585530.5452958553
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
13 1169iT 1 - 169iT
good2 16.25iT32T2 1 - 6.25iT - 32T^{2}
3 1+21.3iT243T2 1 + 21.3iT - 243T^{2}
7 1+116.iT1.68e4T2 1 + 116. iT - 1.68e4T^{2}
11 1+259.T+1.61e5T2 1 + 259.T + 1.61e5T^{2}
17 1+876.iT1.41e6T2 1 + 876. iT - 1.41e6T^{2}
19 1921.T+2.47e6T2 1 - 921.T + 2.47e6T^{2}
23 1+2.05e3iT6.43e6T2 1 + 2.05e3iT - 6.43e6T^{2}
29 1+781.T+2.05e7T2 1 + 781.T + 2.05e7T^{2}
31 1+5.80e3T+2.86e7T2 1 + 5.80e3T + 2.86e7T^{2}
37 12.81e3iT6.93e7T2 1 - 2.81e3iT - 6.93e7T^{2}
41 1+5.40e3T+1.15e8T2 1 + 5.40e3T + 1.15e8T^{2}
43 15.18e3iT1.47e8T2 1 - 5.18e3iT - 1.47e8T^{2}
47 19.66e3iT2.29e8T2 1 - 9.66e3iT - 2.29e8T^{2}
53 1+763.iT4.18e8T2 1 + 763. iT - 4.18e8T^{2}
59 1+4.44e4T+7.14e8T2 1 + 4.44e4T + 7.14e8T^{2}
61 1+5.08e4T+8.44e8T2 1 + 5.08e4T + 8.44e8T^{2}
67 16.73e4iT1.35e9T2 1 - 6.73e4iT - 1.35e9T^{2}
71 1+7.95e4T+1.80e9T2 1 + 7.95e4T + 1.80e9T^{2}
73 1+5.51e4iT2.07e9T2 1 + 5.51e4iT - 2.07e9T^{2}
79 16.64e4T+3.07e9T2 1 - 6.64e4T + 3.07e9T^{2}
83 13.97e4iT3.93e9T2 1 - 3.97e4iT - 3.93e9T^{2}
89 1+5.98e4T+5.58e9T2 1 + 5.98e4T + 5.58e9T^{2}
97 1+4.16e4iT8.58e9T2 1 + 4.16e4iT - 8.58e9T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.46965591068427910740718575786, −9.045738676594913244141566878022, −7.87439955395656975289081220226, −7.42921783309942903585018455991, −6.75149923094796152809336867777, −5.83703545538146828027550499330, −4.65009716430432302613886380304, −2.78728132722754434436462039183, −1.49729564351410859940226877168, −0.13160402507789351749944687784, 1.74781311710972457969136815359, 2.99868734844195749244487282668, 3.76098703836354212517163431541, 4.98317167694896253399841079879, 5.91314089536925104845374446175, 7.52621151723656881452324087281, 8.919081058627703981566489061273, 9.536024327785707647445825287030, 10.43666230240860933508718463175, 10.92260517958753307560318100737

Graph of the ZZ-function along the critical line