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Properties

Label 2-2496-1.1-c3-0-64
Degree 22
Conductor 24962496
Sign 11
Analytic cond. 147.268147.268
Root an. cond. 12.135412.1354
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 + 13.1·5-s + 15.6·7-s + 9·9-s + 55.7·11-s − 13·13-s − 39.3·15-s + 23.4·17-s + 25.6·19-s − 46.8·21-s − 189.·23-s + 47.2·25-s − 27·27-s + 236.·29-s + 47.0·31-s − 167.·33-s + 204.·35-s + 154.·37-s + 39·39-s − 34.6·41-s + 398.·43-s + 118.·45-s + 582.·47-s − 99.1·49-s − 70.4·51-s + 361.·53-s + 731.·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.17·5-s + 0.843·7-s + 0.333·9-s + 1.52·11-s − 0.277·13-s − 0.677·15-s + 0.334·17-s + 0.309·19-s − 0.486·21-s − 1.71·23-s + 0.377·25-s − 0.192·27-s + 1.51·29-s + 0.272·31-s − 0.881·33-s + 0.989·35-s + 0.687·37-s + 0.160·39-s − 0.132·41-s + 1.41·43-s + 0.391·45-s + 1.80·47-s − 0.289·49-s − 0.193·51-s + 0.935·53-s + 1.79·55-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 147.268147.268
Root analytic conductor: 12.135412.1354
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2496, ( :3/2), 1)(2,\ 2496,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.2628938893.262893889
L(12)L(\frac12) ≈\approx 3.2628938893.262893889
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
13 1+13T 1 + 13T
good5 1−13.1T+125T2 1 - 13.1T + 125T^{2}
7 1−15.6T+343T2 1 - 15.6T + 343T^{2}
11 1−55.7T+1.33e3T2 1 - 55.7T + 1.33e3T^{2}
17 1−23.4T+4.91e3T2 1 - 23.4T + 4.91e3T^{2}
19 1−25.6T+6.85e3T2 1 - 25.6T + 6.85e3T^{2}
23 1+189.T+1.21e4T2 1 + 189.T + 1.21e4T^{2}
29 1−236.T+2.43e4T2 1 - 236.T + 2.43e4T^{2}
31 1−47.0T+2.97e4T2 1 - 47.0T + 2.97e4T^{2}
37 1−154.T+5.06e4T2 1 - 154.T + 5.06e4T^{2}
41 1+34.6T+6.89e4T2 1 + 34.6T + 6.89e4T^{2}
43 1−398.T+7.95e4T2 1 - 398.T + 7.95e4T^{2}
47 1−582.T+1.03e5T2 1 - 582.T + 1.03e5T^{2}
53 1−361.T+1.48e5T2 1 - 361.T + 1.48e5T^{2}
59 1+396.T+2.05e5T2 1 + 396.T + 2.05e5T^{2}
61 1−211.T+2.26e5T2 1 - 211.T + 2.26e5T^{2}
67 1+85.8T+3.00e5T2 1 + 85.8T + 3.00e5T^{2}
71 1+651.T+3.57e5T2 1 + 651.T + 3.57e5T^{2}
73 1−927.T+3.89e5T2 1 - 927.T + 3.89e5T^{2}
79 1+1.11e3T+4.93e5T2 1 + 1.11e3T + 4.93e5T^{2}
83 1+391.T+5.71e5T2 1 + 391.T + 5.71e5T^{2}
89 1+745.T+7.04e5T2 1 + 745.T + 7.04e5T^{2}
97 1+173.T+9.12e5T2 1 + 173.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

−8.663040339537159996800988353922, −7.77389835717044415361569491672, −6.88645624154527224327976326832, −6.04468920200788309190695031104, −5.70595912021884655323917946443, −4.61273054487641761704473496898, −3.99933366716651493691274307760, −2.53772379526104580160829602345, −1.62939953893538085647119128250, −0.893979176437509802330674656599, 0.893979176437509802330674656599, 1.62939953893538085647119128250, 2.53772379526104580160829602345, 3.99933366716651493691274307760, 4.61273054487641761704473496898, 5.70595912021884655323917946443, 6.04468920200788309190695031104, 6.88645624154527224327976326832, 7.77389835717044415361569491672, 8.663040339537159996800988353922

Graph of the ZZ-function along the critical line