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Properties

Label 2-88-1.1-c5-0-11
Degree 22
Conductor 8888
Sign −1-1
Analytic cond. 14.113714.1137
Root an. cond. 3.756833.75683
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.16·3-s + 9·5-s − 109.·7-s − 216.·9-s + 121·11-s − 390.·13-s + 46.4·15-s + 422.·17-s − 2.24e3·19-s − 567.·21-s − 217.·23-s − 3.04e3·25-s − 2.37e3·27-s − 4.43e3·29-s + 4.80e3·31-s + 625.·33-s − 988.·35-s − 1.27e4·37-s − 2.01e3·39-s − 7.01e3·41-s + 7.70e3·43-s − 1.94e3·45-s + 1.57e4·47-s − 4.74e3·49-s + 2.18e3·51-s + 1.22e4·53-s + 1.08e3·55-s + ⋯
L(s)  = 1  + 0.331·3-s + 0.160·5-s − 0.847·7-s − 0.890·9-s + 0.301·11-s − 0.640·13-s + 0.0533·15-s + 0.354·17-s − 1.42·19-s − 0.280·21-s − 0.0856·23-s − 0.974·25-s − 0.626·27-s − 0.979·29-s + 0.898·31-s + 0.0999·33-s − 0.136·35-s − 1.53·37-s − 0.212·39-s − 0.651·41-s + 0.635·43-s − 0.143·45-s + 1.04·47-s − 0.282·49-s + 0.117·51-s + 0.599·53-s + 0.0485·55-s + ⋯

Functional equation

Λ(s)=(88s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(88s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8888    =    23⋅112^{3} \cdot 11
Sign: −1-1
Analytic conductor: 14.113714.1137
Root analytic conductor: 3.756833.75683
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 88, ( :5/2), −1)(2,\ 88,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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 1−121T 1 - 121T
good3 1−5.16T+243T2 1 - 5.16T + 243T^{2}
5 1−9T+3.12e3T2 1 - 9T + 3.12e3T^{2}
7 1+109.T+1.68e4T2 1 + 109.T + 1.68e4T^{2}
13 1+390.T+3.71e5T2 1 + 390.T + 3.71e5T^{2}
17 1−422.T+1.41e6T2 1 - 422.T + 1.41e6T^{2}
19 1+2.24e3T+2.47e6T2 1 + 2.24e3T + 2.47e6T^{2}
23 1+217.T+6.43e6T2 1 + 217.T + 6.43e6T^{2}
29 1+4.43e3T+2.05e7T2 1 + 4.43e3T + 2.05e7T^{2}
31 1−4.80e3T+2.86e7T2 1 - 4.80e3T + 2.86e7T^{2}
37 1+1.27e4T+6.93e7T2 1 + 1.27e4T + 6.93e7T^{2}
41 1+7.01e3T+1.15e8T2 1 + 7.01e3T + 1.15e8T^{2}
43 1−7.70e3T+1.47e8T2 1 - 7.70e3T + 1.47e8T^{2}
47 1−1.57e4T+2.29e8T2 1 - 1.57e4T + 2.29e8T^{2}
53 1−1.22e4T+4.18e8T2 1 - 1.22e4T + 4.18e8T^{2}
59 1−2.06e4T+7.14e8T2 1 - 2.06e4T + 7.14e8T^{2}
61 1−2.80e4T+8.44e8T2 1 - 2.80e4T + 8.44e8T^{2}
67 1−5.37e4T+1.35e9T2 1 - 5.37e4T + 1.35e9T^{2}
71 1−2.14e4T+1.80e9T2 1 - 2.14e4T + 1.80e9T^{2}
73 1+2.54e4T+2.07e9T2 1 + 2.54e4T + 2.07e9T^{2}
79 1−2.84e4T+3.07e9T2 1 - 2.84e4T + 3.07e9T^{2}
83 1+6.73e4T+3.93e9T2 1 + 6.73e4T + 3.93e9T^{2}
89 1+3.21e4T+5.58e9T2 1 + 3.21e4T + 5.58e9T^{2}
97 1−1.97e4T+8.58e9T2 1 - 1.97e4T + 8.58e9T^{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

−12.71104090296938544698474588718, −11.68072538132719985891178053807, −10.32919446343358559879481518266, −9.308447901155435725996407038845, −8.243418962414621589370502451663, −6.78275221532505462244127429013, −5.59424691102420022147614481185, −3.77571289706663469526055859700, −2.34191600041547411065664860379, 0, 2.34191600041547411065664860379, 3.77571289706663469526055859700, 5.59424691102420022147614481185, 6.78275221532505462244127429013, 8.243418962414621589370502451663, 9.308447901155435725996407038845, 10.32919446343358559879481518266, 11.68072538132719985891178053807, 12.71104090296938544698474588718

Graph of the ZZ-function along the critical line