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Properties

Label 2-65-1.1-c5-0-19
Degree 22
Conductor 6565
Sign −1-1
Analytic cond. 10.424910.4249
Root an. cond. 3.228763.22876
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 6·3-s − 7·4-s − 25·5-s + 30·6-s − 244·7-s − 195·8-s − 207·9-s − 125·10-s + 794·11-s − 42·12-s − 169·13-s − 1.22e3·14-s − 150·15-s − 751·16-s − 1.53e3·17-s − 1.03e3·18-s + 2.70e3·19-s + 175·20-s − 1.46e3·21-s + 3.97e3·22-s − 702·23-s − 1.17e3·24-s + 625·25-s − 845·26-s − 2.70e3·27-s + 1.70e3·28-s + ⋯
L(s)  = 1  + 0.883·2-s + 0.384·3-s − 0.218·4-s − 0.447·5-s + 0.340·6-s − 1.88·7-s − 1.07·8-s − 0.851·9-s − 0.395·10-s + 1.97·11-s − 0.0841·12-s − 0.277·13-s − 1.66·14-s − 0.172·15-s − 0.733·16-s − 1.28·17-s − 0.752·18-s + 1.71·19-s + 0.0978·20-s − 0.724·21-s + 1.74·22-s − 0.276·23-s − 0.414·24-s + 1/5·25-s − 0.245·26-s − 0.712·27-s + 0.411·28-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 6565    =    5⋅135 \cdot 13
Sign: −1-1
Analytic conductor: 10.424910.4249
Root analytic conductor: 3.228763.22876
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 65, ( :5/2), −1)(2,\ 65,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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+p2T 1 + p^{2} T
13 1+p2T 1 + p^{2} T
good2 1−5T+p5T2 1 - 5 T + p^{5} T^{2}
3 1−2pT+p5T2 1 - 2 p T + p^{5} T^{2}
7 1+244T+p5T2 1 + 244 T + p^{5} T^{2}
11 1−794T+p5T2 1 - 794 T + p^{5} T^{2}
17 1+1534T+p5T2 1 + 1534 T + p^{5} T^{2}
19 1−2706T+p5T2 1 - 2706 T + p^{5} T^{2}
23 1+702T+p5T2 1 + 702 T + p^{5} T^{2}
29 1+5038T+p5T2 1 + 5038 T + p^{5} T^{2}
31 1+3634T+p5T2 1 + 3634 T + p^{5} T^{2}
37 1+7058T+p5T2 1 + 7058 T + p^{5} T^{2}
41 1+294T+p5T2 1 + 294 T + p^{5} T^{2}
43 1−7618T+p5T2 1 - 7618 T + p^{5} T^{2}
47 1+3020T+p5T2 1 + 3020 T + p^{5} T^{2}
53 1−626T+p5T2 1 - 626 T + p^{5} T^{2}
59 1+30066T+p5T2 1 + 30066 T + p^{5} T^{2}
61 1+5806T+p5T2 1 + 5806 T + p^{5} T^{2}
67 1+12436T+p5T2 1 + 12436 T + p^{5} T^{2}
71 1−4734T+p5T2 1 - 4734 T + p^{5} T^{2}
73 1+14694T+p5T2 1 + 14694 T + p^{5} T^{2}
79 1+39804T+p5T2 1 + 39804 T + p^{5} T^{2}
83 1+41776T+p5T2 1 + 41776 T + p^{5} T^{2}
89 1−7970T+p5T2 1 - 7970 T + p^{5} T^{2}
97 1+78050T+p5T2 1 + 78050 T + p^{5} T^{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

−13.48930391804837347601379937064, −12.38030236217803321319551685259, −11.54688191608099333040674511666, −9.401256699229021353461743707036, −9.059771108121277185617321513720, −6.92110022865449866282603655496, −5.84462159905689840514688138944, −3.93818613014492224450947705413, −3.15080588148727670546776940961, 0, 3.15080588148727670546776940961, 3.93818613014492224450947705413, 5.84462159905689840514688138944, 6.92110022865449866282603655496, 9.059771108121277185617321513720, 9.401256699229021353461743707036, 11.54688191608099333040674511666, 12.38030236217803321319551685259, 13.48930391804837347601379937064

Graph of the ZZ-function along the critical line