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Properties

Label 2-77-1.1-c3-0-5
Degree 22
Conductor 7777
Sign −1-1
Analytic cond. 4.543144.54314
Root an. cond. 2.131462.13146
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.05·2-s − 9.64·3-s + 17.5·4-s + 12.9·5-s + 48.7·6-s − 7·7-s − 48.4·8-s + 66.0·9-s − 65.4·10-s + 11·11-s − 169.·12-s − 55.5·13-s + 35.3·14-s − 124.·15-s + 104.·16-s + 59.2·17-s − 333.·18-s − 33.1·19-s + 227.·20-s + 67.5·21-s − 55.6·22-s + 26.0·23-s + 466.·24-s + 42.5·25-s + 280.·26-s − 376.·27-s − 123.·28-s + ⋯
L(s)  = 1  − 1.78·2-s − 1.85·3-s + 2.19·4-s + 1.15·5-s + 3.31·6-s − 0.377·7-s − 2.13·8-s + 2.44·9-s − 2.06·10-s + 0.301·11-s − 4.07·12-s − 1.18·13-s + 0.675·14-s − 2.14·15-s + 1.62·16-s + 0.845·17-s − 4.37·18-s − 0.400·19-s + 2.54·20-s + 0.701·21-s − 0.539·22-s + 0.236·23-s + 3.97·24-s + 0.340·25-s + 2.11·26-s − 2.68·27-s − 0.830·28-s + ⋯

Functional equation

Λ(s)=(77s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(77s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7777    =    7⋅117 \cdot 11
Sign: −1-1
Analytic conductor: 4.543144.54314
Root analytic conductor: 2.131462.13146
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 77, ( :3/2), −1)(2,\ 77,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad7 1+7T 1 + 7T
11 1−11T 1 - 11T
good2 1+5.05T+8T2 1 + 5.05T + 8T^{2}
3 1+9.64T+27T2 1 + 9.64T + 27T^{2}
5 1−12.9T+125T2 1 - 12.9T + 125T^{2}
13 1+55.5T+2.19e3T2 1 + 55.5T + 2.19e3T^{2}
17 1−59.2T+4.91e3T2 1 - 59.2T + 4.91e3T^{2}
19 1+33.1T+6.85e3T2 1 + 33.1T + 6.85e3T^{2}
23 1−26.0T+1.21e4T2 1 - 26.0T + 1.21e4T^{2}
29 1+188.T+2.43e4T2 1 + 188.T + 2.43e4T^{2}
31 1+278.T+2.97e4T2 1 + 278.T + 2.97e4T^{2}
37 1−201.T+5.06e4T2 1 - 201.T + 5.06e4T^{2}
41 1+126.T+6.89e4T2 1 + 126.T + 6.89e4T^{2}
43 1+454.T+7.95e4T2 1 + 454.T + 7.95e4T^{2}
47 1+129.T+1.03e5T2 1 + 129.T + 1.03e5T^{2}
53 1−79.8T+1.48e5T2 1 - 79.8T + 1.48e5T^{2}
59 1+593.T+2.05e5T2 1 + 593.T + 2.05e5T^{2}
61 1+49.8T+2.26e5T2 1 + 49.8T + 2.26e5T^{2}
67 1−295.T+3.00e5T2 1 - 295.T + 3.00e5T^{2}
71 1−546.T+3.57e5T2 1 - 546.T + 3.57e5T^{2}
73 1+809.T+3.89e5T2 1 + 809.T + 3.89e5T^{2}
79 1+375.T+4.93e5T2 1 + 375.T + 4.93e5T^{2}
83 1−85.9T+5.71e5T2 1 - 85.9T + 5.71e5T^{2}
89 1−750.T+7.04e5T2 1 - 750.T + 7.04e5T^{2}
97 1+451.T+9.12e5T2 1 + 451.T + 9.12e5T^{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

−12.92637504752637039576106597423, −11.88337491923734153914492227989, −10.88071735527003557479835441964, −9.967921623863442184276883178611, −9.453554465884656392953467879988, −7.41000213023357838781104720931, −6.44833032670455203553731329261, −5.42217514794295291291569259983, −1.67315106252138622706386683089, 0, 1.67315106252138622706386683089, 5.42217514794295291291569259983, 6.44833032670455203553731329261, 7.41000213023357838781104720931, 9.453554465884656392953467879988, 9.967921623863442184276883178611, 10.88071735527003557479835441964, 11.88337491923734153914492227989, 12.92637504752637039576106597423

Graph of the ZZ-function along the critical line