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Properties

Label 2-768-1.1-c3-0-25
Degree 22
Conductor 768768
Sign 11
Analytic cond. 45.313445.3134
Root an. cond. 6.731526.73152
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 18.4·5-s + 22.4·7-s + 9·9-s + 53.6·11-s − 7.15·13-s − 55.2·15-s + 39.6·17-s + 125.·19-s − 67.2·21-s + 99.1·23-s + 214.·25-s − 27·27-s − 205.·29-s + 147.·31-s − 161.·33-s + 413.·35-s − 125.·37-s + 21.4·39-s − 506.·41-s − 413.·43-s + 165.·45-s − 313.·47-s + 159.·49-s − 119.·51-s + 44.3·53-s + 989.·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.64·5-s + 1.21·7-s + 0.333·9-s + 1.47·11-s − 0.152·13-s − 0.951·15-s + 0.566·17-s + 1.51·19-s − 0.698·21-s + 0.898·23-s + 1.71·25-s − 0.192·27-s − 1.31·29-s + 0.854·31-s − 0.849·33-s + 1.99·35-s − 0.558·37-s + 0.0881·39-s − 1.92·41-s − 1.46·43-s + 0.549·45-s − 0.974·47-s + 0.465·49-s − 0.326·51-s + 0.114·53-s + 2.42·55-s + ⋯

Functional equation

Λ(s)=(768s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(768s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 768768    =    28⋅32^{8} \cdot 3
Sign: 11
Analytic conductor: 45.313445.3134
Root analytic conductor: 6.731526.73152
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 768, ( :3/2), 1)(2,\ 768,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.2258105563.225810556
L(12)L(\frac12) ≈\approx 3.2258105563.225810556
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)
bad2 1 1
3 1+3T 1 + 3T
good5 1−18.4T+125T2 1 - 18.4T + 125T^{2}
7 1−22.4T+343T2 1 - 22.4T + 343T^{2}
11 1−53.6T+1.33e3T2 1 - 53.6T + 1.33e3T^{2}
13 1+7.15T+2.19e3T2 1 + 7.15T + 2.19e3T^{2}
17 1−39.6T+4.91e3T2 1 - 39.6T + 4.91e3T^{2}
19 1−125.T+6.85e3T2 1 - 125.T + 6.85e3T^{2}
23 1−99.1T+1.21e4T2 1 - 99.1T + 1.21e4T^{2}
29 1+205.T+2.43e4T2 1 + 205.T + 2.43e4T^{2}
31 1−147.T+2.97e4T2 1 - 147.T + 2.97e4T^{2}
37 1+125.T+5.06e4T2 1 + 125.T + 5.06e4T^{2}
41 1+506.T+6.89e4T2 1 + 506.T + 6.89e4T^{2}
43 1+413.T+7.95e4T2 1 + 413.T + 7.95e4T^{2}
47 1+313.T+1.03e5T2 1 + 313.T + 1.03e5T^{2}
53 1−44.3T+1.48e5T2 1 - 44.3T + 1.48e5T^{2}
59 1+324T+2.05e5T2 1 + 324T + 2.05e5T^{2}
61 1+324T+2.26e5T2 1 + 324T + 2.26e5T^{2}
67 1+464.T+3.00e5T2 1 + 464.T + 3.00e5T^{2}
71 1−1.05e3T+3.57e5T2 1 - 1.05e3T + 3.57e5T^{2}
73 1−1.02e3T+3.89e5T2 1 - 1.02e3T + 3.89e5T^{2}
79 1+602.T+4.93e5T2 1 + 602.T + 4.93e5T^{2}
83 1+15.8T+5.71e5T2 1 + 15.8T + 5.71e5T^{2}
89 1−381.T+7.04e5T2 1 - 381.T + 7.04e5T^{2}
97 1−659.T+9.12e5T2 1 - 659.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

−9.779448909852847382546060161511, −9.376116940647493922457025597649, −8.303183989998309634106053644143, −7.09610897635939671834875333761, −6.35944992436599034656280145784, −5.33154424284670448438037634895, −4.93416257826124655930273353864, −3.37133580911363636842801467903, −1.73238364787003067350675067632, −1.25834291926636198745532918757, 1.25834291926636198745532918757, 1.73238364787003067350675067632, 3.37133580911363636842801467903, 4.93416257826124655930273353864, 5.33154424284670448438037634895, 6.35944992436599034656280145784, 7.09610897635939671834875333761, 8.303183989998309634106053644143, 9.376116940647493922457025597649, 9.779448909852847382546060161511

Graph of the ZZ-function along the critical line