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Properties

Label 2-325-5.4-c5-0-42
Degree 22
Conductor 325325
Sign 0.447+0.894i0.447 + 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  − 7.50i·2-s + 17.3i·3-s − 24.3·4-s + 130.·6-s + 149. i·7-s − 57.2i·8-s − 58.1·9-s − 492.·11-s − 423. i·12-s − 169i·13-s + 1.12e3·14-s − 1.20e3·16-s − 1.82e3i·17-s + 436. i·18-s − 424.·19-s + ⋯
L(s)  = 1  − 1.32i·2-s + 1.11i·3-s − 0.761·4-s + 1.47·6-s + 1.15i·7-s − 0.316i·8-s − 0.239·9-s − 1.22·11-s − 0.848i·12-s − 0.277i·13-s + 1.53·14-s − 1.18·16-s − 1.53i·17-s + 0.317i·18-s − 0.269·19-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+0.894i0.447 + 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 1.8643193141.864319314
L(12)L(\frac12) ≈\approx 1.8643193141.864319314
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+7.50iT−32T2 1 + 7.50iT - 32T^{2}
3 1−17.3iT−243T2 1 - 17.3iT - 243T^{2}
7 1−149.iT−1.68e4T2 1 - 149. iT - 1.68e4T^{2}
11 1+492.T+1.61e5T2 1 + 492.T + 1.61e5T^{2}
17 1+1.82e3iT−1.41e6T2 1 + 1.82e3iT - 1.41e6T^{2}
19 1+424.T+2.47e6T2 1 + 424.T + 2.47e6T^{2}
23 1+3.66e3iT−6.43e6T2 1 + 3.66e3iT - 6.43e6T^{2}
29 1−8.19e3T+2.05e7T2 1 - 8.19e3T + 2.05e7T^{2}
31 1−8.11e3T+2.86e7T2 1 - 8.11e3T + 2.86e7T^{2}
37 1−6.29e3iT−6.93e7T2 1 - 6.29e3iT - 6.93e7T^{2}
41 1−4.29e3T+1.15e8T2 1 - 4.29e3T + 1.15e8T^{2}
43 1−5.82e3iT−1.47e8T2 1 - 5.82e3iT - 1.47e8T^{2}
47 1−8.50e3iT−2.29e8T2 1 - 8.50e3iT - 2.29e8T^{2}
53 1−7.04e3iT−4.18e8T2 1 - 7.04e3iT - 4.18e8T^{2}
59 1−2.15e4T+7.14e8T2 1 - 2.15e4T + 7.14e8T^{2}
61 1−3.57e4T+8.44e8T2 1 - 3.57e4T + 8.44e8T^{2}
67 1+1.52e4iT−1.35e9T2 1 + 1.52e4iT - 1.35e9T^{2}
71 1−6.91e4T+1.80e9T2 1 - 6.91e4T + 1.80e9T^{2}
73 1+4.13e3iT−2.07e9T2 1 + 4.13e3iT - 2.07e9T^{2}
79 1−1.60e4T+3.07e9T2 1 - 1.60e4T + 3.07e9T^{2}
83 1+1.15e5iT−3.93e9T2 1 + 1.15e5iT - 3.93e9T^{2}
89 1−3.52e4T+5.58e9T2 1 - 3.52e4T + 5.58e9T^{2}
97 1+3.21e4iT−8.58e9T2 1 + 3.21e4iT - 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

−10.45201687758172883401365643497, −10.04233128099490581992637466830, −9.147674734229194011848924846073, −8.214324291834379192389817057462, −6.57810878930727093234539995492, −5.08839984411517801036699204944, −4.49005051830412697256678397686, −2.89879408365057798961170212146, −2.55981906642301570068162369901, −0.64512586621269880656358124284, 0.874186170545733273698688366043, 2.25651110159700338377074756800, 4.09476344328708541469434066846, 5.35988097830303633545635542894, 6.45868743625371950490454349312, 7.02994233302115202859974909201, 7.945368557020319902541100752235, 8.303841858885461398810312915954, 10.00838926192082830864333821053, 10.85092580386724506624407795430

Graph of the ZZ-function along the critical line