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Properties

Label 2-325-5.4-c5-0-68
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  − 5.84i·2-s + 12.3i·3-s − 2.10·4-s + 72.1·6-s − 135. i·7-s − 174. i·8-s + 90.3·9-s + 564.·11-s − 26.0i·12-s − 169i·13-s − 788.·14-s − 1.08e3·16-s − 1.44e3i·17-s − 527. i·18-s + 530.·19-s + ⋯
L(s)  = 1  − 1.03i·2-s + 0.792i·3-s − 0.0658·4-s + 0.818·6-s − 1.04i·7-s − 0.964i·8-s + 0.371·9-s + 1.40·11-s − 0.0521i·12-s − 0.277i·13-s − 1.07·14-s − 1.06·16-s − 1.21i·17-s − 0.383i·18-s + 0.337·19-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 2.5650745702.565074570
L(12)L(\frac12) ≈\approx 2.5650745702.565074570
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+5.84iT−32T2 1 + 5.84iT - 32T^{2}
3 1−12.3iT−243T2 1 - 12.3iT - 243T^{2}
7 1+135.iT−1.68e4T2 1 + 135. iT - 1.68e4T^{2}
11 1−564.T+1.61e5T2 1 - 564.T + 1.61e5T^{2}
17 1+1.44e3iT−1.41e6T2 1 + 1.44e3iT - 1.41e6T^{2}
19 1−530.T+2.47e6T2 1 - 530.T + 2.47e6T^{2}
23 1−4.71e3iT−6.43e6T2 1 - 4.71e3iT - 6.43e6T^{2}
29 1+3.32e3T+2.05e7T2 1 + 3.32e3T + 2.05e7T^{2}
31 1−1.86e3T+2.86e7T2 1 - 1.86e3T + 2.86e7T^{2}
37 1+1.02e4iT−6.93e7T2 1 + 1.02e4iT - 6.93e7T^{2}
41 1+1.79e4T+1.15e8T2 1 + 1.79e4T + 1.15e8T^{2}
43 1−7.56e3iT−1.47e8T2 1 - 7.56e3iT - 1.47e8T^{2}
47 1+2.40e4iT−2.29e8T2 1 + 2.40e4iT - 2.29e8T^{2}
53 1+5.49e3iT−4.18e8T2 1 + 5.49e3iT - 4.18e8T^{2}
59 1−4.54e4T+7.14e8T2 1 - 4.54e4T + 7.14e8T^{2}
61 1−2.93e4T+8.44e8T2 1 - 2.93e4T + 8.44e8T^{2}
67 1+6.99e4iT−1.35e9T2 1 + 6.99e4iT - 1.35e9T^{2}
71 1−4.91e4T+1.80e9T2 1 - 4.91e4T + 1.80e9T^{2}
73 1+4.13e4iT−2.07e9T2 1 + 4.13e4iT - 2.07e9T^{2}
79 1+4.84e4T+3.07e9T2 1 + 4.84e4T + 3.07e9T^{2}
83 1+1.06e5iT−3.93e9T2 1 + 1.06e5iT - 3.93e9T^{2}
89 1−2.33e4T+5.58e9T2 1 - 2.33e4T + 5.58e9T^{2}
97 1−7.75e4iT−8.58e9T2 1 - 7.75e4iT - 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

−10.46110039864845255293462895732, −9.730267574420325740589210375175, −9.218924549307231840933372186383, −7.42288592828611569139328357492, −6.79071395632317220731356604531, −5.12575520538648091855957770060, −3.84411263116080442237939057130, −3.51378286914115127078644807307, −1.72919836255097088158160211516, −0.70166395548283325072369679538, 1.37856277933832828852985977666, 2.40642332623097353548037067330, 4.20250611857900345369555922285, 5.58792442969632903347865325850, 6.54695905359745573786327372048, 6.86827011961163260066104223822, 8.251932034365359853323552101508, 8.685109261953246114143489033067, 9.976513830803770978630269207724, 11.35823990547219859818358347985

Graph of the ZZ-function along the critical line