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Properties

Label 2-65-5.4-c5-0-22
Degree 22
Conductor 6565
Sign 0.338+0.940i0.338 + 0.940i
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  + 3.30i·2-s − 24.0i·3-s + 21.1·4-s + (52.5 − 18.9i)5-s + 79.3·6-s − 91.7i·7-s + 175. i·8-s − 335.·9-s + (62.5 + 173. i)10-s − 45.5·11-s − 507. i·12-s − 169i·13-s + 302.·14-s + (−455. − 1.26e3i)15-s + 96.5·16-s − 745. i·17-s + ⋯
L(s)  = 1  + 0.583i·2-s − 1.54i·3-s + 0.659·4-s + (0.940 − 0.338i)5-s + 0.900·6-s − 0.707i·7-s + 0.968i·8-s − 1.37·9-s + (0.197 + 0.549i)10-s − 0.113·11-s − 1.01i·12-s − 0.277i·13-s + 0.412·14-s + (−0.522 − 1.45i)15-s + 0.0942·16-s − 0.625i·17-s + ⋯

Functional equation

Λ(s)=(65s/2 ΓC(s) L(s)=((0.338+0.940i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(65s/2 ΓC(s+5/2) L(s)=((0.338+0.940i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6565    =    5⋅135 \cdot 13
Sign: 0.338+0.940i0.338 + 0.940i
Analytic conductor: 10.424910.4249
Root analytic conductor: 3.228763.22876
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ65(14,⋅)\chi_{65} (14, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 65, ( :5/2), 0.338+0.940i)(2,\ 65,\ (\ :5/2),\ 0.338 + 0.940i)

Particular Values

L(3)L(3) ≈\approx 1.88070−1.32171i1.88070 - 1.32171i
L(12)L(\frac12) ≈\approx 1.88070−1.32171i1.88070 - 1.32171i
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+(−52.5+18.9i)T 1 + (-52.5 + 18.9i)T
13 1+169iT 1 + 169iT
good2 1−3.30iT−32T2 1 - 3.30iT - 32T^{2}
3 1+24.0iT−243T2 1 + 24.0iT - 243T^{2}
7 1+91.7iT−1.68e4T2 1 + 91.7iT - 1.68e4T^{2}
11 1+45.5T+1.61e5T2 1 + 45.5T + 1.61e5T^{2}
17 1+745.iT−1.41e6T2 1 + 745. iT - 1.41e6T^{2}
19 1+306.T+2.47e6T2 1 + 306.T + 2.47e6T^{2}
23 1−704.iT−6.43e6T2 1 - 704. iT - 6.43e6T^{2}
29 1−6.80e3T+2.05e7T2 1 - 6.80e3T + 2.05e7T^{2}
31 1+7.60e3T+2.86e7T2 1 + 7.60e3T + 2.86e7T^{2}
37 1−109.iT−6.93e7T2 1 - 109. iT - 6.93e7T^{2}
41 1+1.33e4T+1.15e8T2 1 + 1.33e4T + 1.15e8T^{2}
43 1−4.47e3iT−1.47e8T2 1 - 4.47e3iT - 1.47e8T^{2}
47 1+4.86e3iT−2.29e8T2 1 + 4.86e3iT - 2.29e8T^{2}
53 1−4.01e4iT−4.18e8T2 1 - 4.01e4iT - 4.18e8T^{2}
59 1−2.96e4T+7.14e8T2 1 - 2.96e4T + 7.14e8T^{2}
61 1−5.15e4T+8.44e8T2 1 - 5.15e4T + 8.44e8T^{2}
67 1−5.18e4iT−1.35e9T2 1 - 5.18e4iT - 1.35e9T^{2}
71 1−4.22e4T+1.80e9T2 1 - 4.22e4T + 1.80e9T^{2}
73 1−4.52e4iT−2.07e9T2 1 - 4.52e4iT - 2.07e9T^{2}
79 1+6.25e4T+3.07e9T2 1 + 6.25e4T + 3.07e9T^{2}
83 1+4.69e4iT−3.93e9T2 1 + 4.69e4iT - 3.93e9T^{2}
89 1−6.33e4T+5.58e9T2 1 - 6.33e4T + 5.58e9T^{2}
97 1−1.32e5iT−8.58e9T2 1 - 1.32e5iT - 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

−13.67083131817975682056554933462, −12.78559432997700251349324086692, −11.67795533031609564678502539496, −10.31702123562856424697504309485, −8.530202437404375390155281737468, −7.35663243721989227896902237248, −6.60637804230525373608756842810, −5.45567447632637290041855342874, −2.44756341579527529343377150973, −1.12718699783522020152161732507, 2.18026151591968824343638954773, 3.53439030119662094281475554339, 5.25798585043833352305865618742, 6.54131517139774429621699003661, 8.784416684000949074671030315493, 9.911131026899539162507751133285, 10.51410751728417083166941049063, 11.51788205285818736904899502547, 12.82533289009337421349613624747, 14.41282767779183503389903759603

Graph of the ZZ-function along the critical line