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Properties

Label 2-325-65.7-c1-0-13
Degree 22
Conductor 325325
Sign 0.581+0.813i0.581 + 0.813i
Analytic cond. 2.595132.59513
Root an. cond. 1.610941.61094
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.427 + 0.246i)2-s + (0.243 + 0.908i)3-s + (−0.878 − 1.52i)4-s + (−0.120 + 0.448i)6-s + (−1.83 − 3.18i)7-s − 1.85i·8-s + (1.83 − 1.05i)9-s + (−0.177 − 0.664i)11-s + (1.16 − 1.16i)12-s + (2.92 − 2.11i)13-s − 1.81i·14-s + (−1.29 + 2.24i)16-s + (−2.29 − 0.614i)17-s + 1.04·18-s + (5.29 + 1.41i)19-s + ⋯
L(s)  = 1  + (0.302 + 0.174i)2-s + (0.140 + 0.524i)3-s + (−0.439 − 0.760i)4-s + (−0.0490 + 0.183i)6-s + (−0.694 − 1.20i)7-s − 0.655i·8-s + (0.610 − 0.352i)9-s + (−0.0536 − 0.200i)11-s + (0.337 − 0.337i)12-s + (0.810 − 0.585i)13-s − 0.485i·14-s + (−0.324 + 0.562i)16-s + (−0.556 − 0.149i)17-s + 0.246·18-s + (1.21 + 0.325i)19-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((0.581+0.813i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+1/2) L(s)=((0.581+0.813i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: 0.581+0.813i0.581 + 0.813i
Analytic conductor: 2.595132.59513
Root analytic conductor: 1.610941.61094
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ325(7,⋅)\chi_{325} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :1/2), 0.581+0.813i)(2,\ 325,\ (\ :1/2),\ 0.581 + 0.813i)

Particular Values

L(1)L(1) ≈\approx 1.20794−0.620960i1.20794 - 0.620960i
L(12)L(\frac12) ≈\approx 1.20794−0.620960i1.20794 - 0.620960i
L(32)L(\frac{3}{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+(−2.92+2.11i)T 1 + (-2.92 + 2.11i)T
good2 1+(−0.427−0.246i)T+(1+1.73i)T2 1 + (-0.427 - 0.246i)T + (1 + 1.73i)T^{2}
3 1+(−0.243−0.908i)T+(−2.59+1.5i)T2 1 + (-0.243 - 0.908i)T + (-2.59 + 1.5i)T^{2}
7 1+(1.83+3.18i)T+(−3.5+6.06i)T2 1 + (1.83 + 3.18i)T + (-3.5 + 6.06i)T^{2}
11 1+(0.177+0.664i)T+(−9.52+5.5i)T2 1 + (0.177 + 0.664i)T + (-9.52 + 5.5i)T^{2}
17 1+(2.29+0.614i)T+(14.7+8.5i)T2 1 + (2.29 + 0.614i)T + (14.7 + 8.5i)T^{2}
19 1+(−5.29−1.41i)T+(16.4+9.5i)T2 1 + (-5.29 - 1.41i)T + (16.4 + 9.5i)T^{2}
23 1+(1.30−0.350i)T+(19.9−11.5i)T2 1 + (1.30 - 0.350i)T + (19.9 - 11.5i)T^{2}
29 1+(8.24+4.75i)T+(14.5+25.1i)T2 1 + (8.24 + 4.75i)T + (14.5 + 25.1i)T^{2}
31 1+(−4.81−4.81i)T+31iT2 1 + (-4.81 - 4.81i)T + 31iT^{2}
37 1+(−0.917+1.58i)T+(−18.5−32.0i)T2 1 + (-0.917 + 1.58i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.534−0.143i)T+(35.5−20.5i)T2 1 + (0.534 - 0.143i)T + (35.5 - 20.5i)T^{2}
43 1+(−0.560+2.09i)T+(−37.2−21.5i)T2 1 + (-0.560 + 2.09i)T + (-37.2 - 21.5i)T^{2}
47 1−3.80T+47T2 1 - 3.80T + 47T^{2}
53 1+(−2.47+2.47i)T−53iT2 1 + (-2.47 + 2.47i)T - 53iT^{2}
59 1+(2.69−10.0i)T+(−51.0−29.5i)T2 1 + (2.69 - 10.0i)T + (-51.0 - 29.5i)T^{2}
61 1+(3.09+5.36i)T+(−30.5+52.8i)T2 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2}
67 1+(−10.6−6.12i)T+(33.5+58.0i)T2 1 + (-10.6 - 6.12i)T + (33.5 + 58.0i)T^{2}
71 1+(1.73−6.47i)T+(−61.4−35.5i)T2 1 + (1.73 - 6.47i)T + (-61.4 - 35.5i)T^{2}
73 1−3.37iT−73T2 1 - 3.37iT - 73T^{2}
79 1−3.12iT−79T2 1 - 3.12iT - 79T^{2}
83 1+2.13T+83T2 1 + 2.13T + 83T^{2}
89 1+(−3.26+0.874i)T+(77.0−44.5i)T2 1 + (-3.26 + 0.874i)T + (77.0 - 44.5i)T^{2}
97 1+(6.12−3.53i)T+(48.5−84.0i)T2 1 + (6.12 - 3.53i)T + (48.5 - 84.0i)T^{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

−11.21543121016208178353298920190, −10.26312245168278901460571579126, −9.848965089236430535609223631181, −8.913134170170375703964970524100, −7.46741022991755287527303788953, −6.51166930067263998762649831241, −5.46738826280203900016216197689, −4.19116924730912733967428953677, −3.53778010702335621843416175318, −0.954342106547333236551374524965, 2.12185049125853873392080784039, 3.33279679663753754717103605545, 4.59510289801482437736101907364, 5.84255005779189802038537117508, 7.00249680182345525595353330237, 7.979941841605551712037314085082, 8.957069536107144198464628482929, 9.610579217596306882709214693745, 11.14385957345830377419776971200, 11.98679301891509976518963093496

Graph of the ZZ-function along the critical line