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Properties

Label 2-65-13.12-c5-0-9
Degree 22
Conductor 6565
Sign 0.0811+0.996i0.0811 + 0.996i
Analytic cond. 10.424910.4249
Root an. cond. 3.228763.22876
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  − 3.56i·2-s − 28.5·3-s + 19.2·4-s + 25i·5-s + 101. i·6-s + 143. i·7-s − 182. i·8-s + 572.·9-s + 89.1·10-s − 392. i·11-s − 550.·12-s + (−607. + 49.4i)13-s + 512.·14-s − 714. i·15-s − 35.9·16-s + 484.·17-s + ⋯
L(s)  = 1  − 0.630i·2-s − 1.83·3-s + 0.602·4-s + 0.447i·5-s + 1.15i·6-s + 1.10i·7-s − 1.01i·8-s + 2.35·9-s + 0.282·10-s − 0.976i·11-s − 1.10·12-s + (−0.996 + 0.0811i)13-s + 0.698·14-s − 0.819i·15-s − 0.0350·16-s + 0.406·17-s + ⋯

Functional equation

Λ(s)=(65s/2 ΓC(s) L(s)=((0.0811+0.996i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0811 + 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(65s/2 ΓC(s+5/2) L(s)=((0.0811+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0811 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6565    =    5⋅135 \cdot 13
Sign: 0.0811+0.996i0.0811 + 0.996i
Analytic conductor: 10.424910.4249
Root analytic conductor: 3.228763.22876
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ65(51,⋅)\chi_{65} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 65, ( :5/2), 0.0811+0.996i)(2,\ 65,\ (\ :5/2),\ 0.0811 + 0.996i)

Particular Values

L(3)L(3) ≈\approx 0.729447−0.672485i0.729447 - 0.672485i
L(12)L(\frac12) ≈\approx 0.729447−0.672485i0.729447 - 0.672485i
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 1−25iT 1 - 25iT
13 1+(607.−49.4i)T 1 + (607. - 49.4i)T
good2 1+3.56iT−32T2 1 + 3.56iT - 32T^{2}
3 1+28.5T+243T2 1 + 28.5T + 243T^{2}
7 1−143.iT−1.68e4T2 1 - 143. iT - 1.68e4T^{2}
11 1+392.iT−1.61e5T2 1 + 392. iT - 1.61e5T^{2}
17 1−484.T+1.41e6T2 1 - 484.T + 1.41e6T^{2}
19 1+3.07e3iT−2.47e6T2 1 + 3.07e3iT - 2.47e6T^{2}
23 1−3.91e3T+6.43e6T2 1 - 3.91e3T + 6.43e6T^{2}
29 1−4.31e3T+2.05e7T2 1 - 4.31e3T + 2.05e7T^{2}
31 1+2.95e3iT−2.86e7T2 1 + 2.95e3iT - 2.86e7T^{2}
37 1+6.95e3iT−6.93e7T2 1 + 6.95e3iT - 6.93e7T^{2}
41 1−1.63e4iT−1.15e8T2 1 - 1.63e4iT - 1.15e8T^{2}
43 1−1.56e4T+1.47e8T2 1 - 1.56e4T + 1.47e8T^{2}
47 1+1.52e4iT−2.29e8T2 1 + 1.52e4iT - 2.29e8T^{2}
53 1+1.04e4T+4.18e8T2 1 + 1.04e4T + 4.18e8T^{2}
59 1+2.08e4iT−7.14e8T2 1 + 2.08e4iT - 7.14e8T^{2}
61 1+650.T+8.44e8T2 1 + 650.T + 8.44e8T^{2}
67 1+9.52e3iT−1.35e9T2 1 + 9.52e3iT - 1.35e9T^{2}
71 1−1.53e4iT−1.80e9T2 1 - 1.53e4iT - 1.80e9T^{2}
73 1+4.98e4iT−2.07e9T2 1 + 4.98e4iT - 2.07e9T^{2}
79 1+5.91e3T+3.07e9T2 1 + 5.91e3T + 3.07e9T^{2}
83 1+1.78e4iT−3.93e9T2 1 + 1.78e4iT - 3.93e9T^{2}
89 1−9.27e4iT−5.58e9T2 1 - 9.27e4iT - 5.58e9T^{2}
97 1+3.30e4iT−8.58e9T2 1 + 3.30e4iT - 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

−13.02307395810763118012928148724, −12.09894845688506541245615179857, −11.34415961622251389746989072317, −10.78771462007950654172274366113, −9.458327033090688109435339577564, −7.12226933031964304339881483467, −6.18433517971415813905193347345, −5.00935273626032846830103082247, −2.70054203218576788367006441992, −0.66146591467384761655772876858, 1.23280205251044607049716044386, 4.56143890372930453828251274180, 5.60285353023722427607784776410, 6.87597668768682448447969254480, 7.57721839869209859025319638532, 10.04720602904560664608672188913, 10.73893237542796609682795234832, 12.04625282686386073219512661077, 12.57255773972694059115438343119, 14.36965014530062324846346183411

Graph of the ZZ-function along the critical line