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Properties

Label 2-325-5.4-c5-0-63
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  − 3.78i·2-s − 8.20i·3-s + 17.6·4-s − 31.0·6-s + 88.9i·7-s − 188. i·8-s + 175.·9-s − 156.·11-s − 145. i·12-s − 169i·13-s + 336.·14-s − 145.·16-s + 447. i·17-s − 664. i·18-s + 269.·19-s + ⋯
L(s)  = 1  − 0.668i·2-s − 0.526i·3-s + 0.552·4-s − 0.352·6-s + 0.686i·7-s − 1.03i·8-s + 0.722·9-s − 0.390·11-s − 0.290i·12-s − 0.277i·13-s + 0.459·14-s − 0.142·16-s + 0.375i·17-s − 0.483i·18-s + 0.171·19-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 2.6250255742.625025574
L(12)L(\frac12) ≈\approx 2.6250255742.625025574
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+3.78iT−32T2 1 + 3.78iT - 32T^{2}
3 1+8.20iT−243T2 1 + 8.20iT - 243T^{2}
7 1−88.9iT−1.68e4T2 1 - 88.9iT - 1.68e4T^{2}
11 1+156.T+1.61e5T2 1 + 156.T + 1.61e5T^{2}
17 1−447.iT−1.41e6T2 1 - 447. iT - 1.41e6T^{2}
19 1−269.T+2.47e6T2 1 - 269.T + 2.47e6T^{2}
23 1+1.37e3iT−6.43e6T2 1 + 1.37e3iT - 6.43e6T^{2}
29 1−3.69e3T+2.05e7T2 1 - 3.69e3T + 2.05e7T^{2}
31 1−797.T+2.86e7T2 1 - 797.T + 2.86e7T^{2}
37 1+4.39e3iT−6.93e7T2 1 + 4.39e3iT - 6.93e7T^{2}
41 1−1.43e4T+1.15e8T2 1 - 1.43e4T + 1.15e8T^{2}
43 1+1.13e4iT−1.47e8T2 1 + 1.13e4iT - 1.47e8T^{2}
47 1+9.97e3iT−2.29e8T2 1 + 9.97e3iT - 2.29e8T^{2}
53 1+1.15e4iT−4.18e8T2 1 + 1.15e4iT - 4.18e8T^{2}
59 1+7.13e3T+7.14e8T2 1 + 7.13e3T + 7.14e8T^{2}
61 1−1.66e4T+8.44e8T2 1 - 1.66e4T + 8.44e8T^{2}
67 1−4.21e3iT−1.35e9T2 1 - 4.21e3iT - 1.35e9T^{2}
71 1−1.28e4T+1.80e9T2 1 - 1.28e4T + 1.80e9T^{2}
73 1−1.30e4iT−2.07e9T2 1 - 1.30e4iT - 2.07e9T^{2}
79 1+7.47e4T+3.07e9T2 1 + 7.47e4T + 3.07e9T^{2}
83 1+8.54e3iT−3.93e9T2 1 + 8.54e3iT - 3.93e9T^{2}
89 1+5.95e4T+5.58e9T2 1 + 5.95e4T + 5.58e9T^{2}
97 1+1.73e4iT−8.58e9T2 1 + 1.73e4iT - 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

−10.52170641979929578500788896811, −9.833681550909324982747309189538, −8.583251504434373456831021302036, −7.51285482202065913064276508399, −6.66448648677402000510918547861, −5.64230680119232670955078797939, −4.14834568141101641153408609077, −2.79319066634405293823635598369, −1.92290755612145368656059573563, −0.72195375650107180667651601060, 1.24971763256407764072469099259, 2.77706767818221027446723659999, 4.17315358991876740513198613666, 5.15543729445058983672468479508, 6.35760708581194387992497164088, 7.26330665869461945153923952714, 7.932439714749123117678739107948, 9.244983650484167423809243783536, 10.21282391674200342950759019979, 10.92478876659734288405772046629

Graph of the ZZ-function along the critical line