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Properties

Label 2-6e4-1.1-c3-0-11
Degree 22
Conductor 12961296
Sign 11
Analytic cond. 76.466476.4664
Root an. cond. 8.744518.74451
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  + 13.8·5-s − 30.7·7-s − 43.9·11-s − 12.2·13-s − 76.0·17-s + 44.1·19-s − 78.6·23-s + 66.6·25-s + 92.7·29-s + 143.·31-s − 425.·35-s − 32.4·37-s + 335.·41-s + 498.·43-s − 281.·47-s + 599.·49-s + 628.·53-s − 607.·55-s + 504.·59-s + 371.·61-s − 169.·65-s + 162.·67-s + 433.·71-s − 629.·73-s + 1.34e3·77-s − 172.·79-s − 174.·83-s + ⋯
L(s)  = 1  + 1.23·5-s − 1.65·7-s − 1.20·11-s − 0.261·13-s − 1.08·17-s + 0.533·19-s − 0.712·23-s + 0.533·25-s + 0.594·29-s + 0.828·31-s − 2.05·35-s − 0.144·37-s + 1.27·41-s + 1.76·43-s − 0.874·47-s + 1.74·49-s + 1.62·53-s − 1.49·55-s + 1.11·59-s + 0.780·61-s − 0.323·65-s + 0.296·67-s + 0.724·71-s − 1.00·73-s + 1.99·77-s − 0.245·79-s − 0.231·83-s + ⋯

Functional equation

Λ(s)=(1296s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1296s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12961296    =    24⋅342^{4} \cdot 3^{4}
Sign: 11
Analytic conductor: 76.466476.4664
Root analytic conductor: 8.744518.74451
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1296, ( :3/2), 1)(2,\ 1296,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.5888362001.588836200
L(12)L(\frac12) ≈\approx 1.5888362001.588836200
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 1
good5 1−13.8T+125T2 1 - 13.8T + 125T^{2}
7 1+30.7T+343T2 1 + 30.7T + 343T^{2}
11 1+43.9T+1.33e3T2 1 + 43.9T + 1.33e3T^{2}
13 1+12.2T+2.19e3T2 1 + 12.2T + 2.19e3T^{2}
17 1+76.0T+4.91e3T2 1 + 76.0T + 4.91e3T^{2}
19 1−44.1T+6.85e3T2 1 - 44.1T + 6.85e3T^{2}
23 1+78.6T+1.21e4T2 1 + 78.6T + 1.21e4T^{2}
29 1−92.7T+2.43e4T2 1 - 92.7T + 2.43e4T^{2}
31 1−143.T+2.97e4T2 1 - 143.T + 2.97e4T^{2}
37 1+32.4T+5.06e4T2 1 + 32.4T + 5.06e4T^{2}
41 1−335.T+6.89e4T2 1 - 335.T + 6.89e4T^{2}
43 1−498.T+7.95e4T2 1 - 498.T + 7.95e4T^{2}
47 1+281.T+1.03e5T2 1 + 281.T + 1.03e5T^{2}
53 1−628.T+1.48e5T2 1 - 628.T + 1.48e5T^{2}
59 1−504.T+2.05e5T2 1 - 504.T + 2.05e5T^{2}
61 1−371.T+2.26e5T2 1 - 371.T + 2.26e5T^{2}
67 1−162.T+3.00e5T2 1 - 162.T + 3.00e5T^{2}
71 1−433.T+3.57e5T2 1 - 433.T + 3.57e5T^{2}
73 1+629.T+3.89e5T2 1 + 629.T + 3.89e5T^{2}
79 1+172.T+4.93e5T2 1 + 172.T + 4.93e5T^{2}
83 1+174.T+5.71e5T2 1 + 174.T + 5.71e5T^{2}
89 1+336.T+7.04e5T2 1 + 336.T + 7.04e5T^{2}
97 1−84.3T+9.12e5T2 1 - 84.3T + 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

−9.458364753714543385855944255958, −8.684908288836385399958304489717, −7.53493915166460502041752421523, −6.63476939595013231607871210030, −6.00416098783160082425303248713, −5.29925931328052870897667256034, −4.07998775538846882365237378592, −2.77667357553586694059405334440, −2.31857454177097551538411424767, −0.60149297450461525633310492309, 0.60149297450461525633310492309, 2.31857454177097551538411424767, 2.77667357553586694059405334440, 4.07998775538846882365237378592, 5.29925931328052870897667256034, 6.00416098783160082425303248713, 6.63476939595013231607871210030, 7.53493915166460502041752421523, 8.684908288836385399958304489717, 9.458364753714543385855944255958

Graph of the ZZ-function along the critical line