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Properties

Label 2-88-1.1-c3-0-2
Degree 22
Conductor 8888
Sign 11
Analytic cond. 5.192165.19216
Root an. cond. 2.278632.27863
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  + 8.57·3-s − 8.24·5-s + 26.8·7-s + 46.5·9-s + 11·11-s − 72.1·13-s − 70.7·15-s + 116.·17-s − 70.6·19-s + 230.·21-s + 21.2·23-s − 57.0·25-s + 167.·27-s + 38.4·29-s − 263.·31-s + 94.3·33-s − 221.·35-s − 156.·37-s − 618.·39-s − 112.·41-s + 59.7·43-s − 383.·45-s − 134.·47-s + 376.·49-s + 1.00e3·51-s − 585.·53-s − 90.6·55-s + ⋯
L(s)  = 1  + 1.65·3-s − 0.737·5-s + 1.44·7-s + 1.72·9-s + 0.301·11-s − 1.53·13-s − 1.21·15-s + 1.66·17-s − 0.852·19-s + 2.39·21-s + 0.192·23-s − 0.456·25-s + 1.19·27-s + 0.246·29-s − 1.52·31-s + 0.497·33-s − 1.06·35-s − 0.694·37-s − 2.53·39-s − 0.426·41-s + 0.211·43-s − 1.27·45-s − 0.417·47-s + 1.09·49-s + 2.74·51-s − 1.51·53-s − 0.222·55-s + ⋯

Functional equation

Λ(s)=(88s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(88s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8888    =    23⋅112^{3} \cdot 11
Sign: 11
Analytic conductor: 5.192165.19216
Root analytic conductor: 2.278632.27863
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 88, ( :3/2), 1)(2,\ 88,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.4537703972.453770397
L(12)L(\frac12) ≈\approx 2.4537703972.453770397
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
11 1−11T 1 - 11T
good3 1−8.57T+27T2 1 - 8.57T + 27T^{2}
5 1+8.24T+125T2 1 + 8.24T + 125T^{2}
7 1−26.8T+343T2 1 - 26.8T + 343T^{2}
13 1+72.1T+2.19e3T2 1 + 72.1T + 2.19e3T^{2}
17 1−116.T+4.91e3T2 1 - 116.T + 4.91e3T^{2}
19 1+70.6T+6.85e3T2 1 + 70.6T + 6.85e3T^{2}
23 1−21.2T+1.21e4T2 1 - 21.2T + 1.21e4T^{2}
29 1−38.4T+2.43e4T2 1 - 38.4T + 2.43e4T^{2}
31 1+263.T+2.97e4T2 1 + 263.T + 2.97e4T^{2}
37 1+156.T+5.06e4T2 1 + 156.T + 5.06e4T^{2}
41 1+112.T+6.89e4T2 1 + 112.T + 6.89e4T^{2}
43 1−59.7T+7.95e4T2 1 - 59.7T + 7.95e4T^{2}
47 1+134.T+1.03e5T2 1 + 134.T + 1.03e5T^{2}
53 1+585.T+1.48e5T2 1 + 585.T + 1.48e5T^{2}
59 1−573.T+2.05e5T2 1 - 573.T + 2.05e5T^{2}
61 1−347.T+2.26e5T2 1 - 347.T + 2.26e5T^{2}
67 1+1.05e3T+3.00e5T2 1 + 1.05e3T + 3.00e5T^{2}
71 1+292.T+3.57e5T2 1 + 292.T + 3.57e5T^{2}
73 1−230.T+3.89e5T2 1 - 230.T + 3.89e5T^{2}
79 1−1.16e3T+4.93e5T2 1 - 1.16e3T + 4.93e5T^{2}
83 1−763.T+5.71e5T2 1 - 763.T + 5.71e5T^{2}
89 1−1.02e3T+7.04e5T2 1 - 1.02e3T + 7.04e5T^{2}
97 1−832.T+9.12e5T2 1 - 832.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

−14.15112838371468489555260026407, −12.64555432174283018800162517979, −11.67000859361383516785327882692, −10.19190512681691691654648637078, −8.982482989224976186176405820413, −7.910315789992655858607359498043, −7.47839850254553866370060509211, −4.88537701709424161140493738813, −3.56759448675284572070662362310, −1.95346657413988652683036824863, 1.95346657413988652683036824863, 3.56759448675284572070662362310, 4.88537701709424161140493738813, 7.47839850254553866370060509211, 7.910315789992655858607359498043, 8.982482989224976186176405820413, 10.19190512681691691654648637078, 11.67000859361383516785327882692, 12.64555432174283018800162517979, 14.15112838371468489555260026407

Graph of the ZZ-function along the critical line