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Properties

Label 2-1160-1.1-c1-0-18
Degree 22
Conductor 11601160
Sign −1-1
Analytic cond. 9.262649.26264
Root an. cond. 3.043453.04345
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.90·3-s + 5-s + 0.903·7-s + 5.42·9-s − 5.52·11-s − 0.622·13-s − 2.90·15-s + 3.52·17-s − 1.09·19-s − 2.62·21-s + 5.33·23-s + 25-s − 7.05·27-s − 29-s − 1.65·31-s + 16.0·33-s + 0.903·35-s − 2.28·37-s + 1.80·39-s − 7.67·41-s − 1.09·43-s + 5.42·45-s − 1.65·47-s − 6.18·49-s − 10.2·51-s − 2.42·53-s − 5.52·55-s + ⋯
L(s)  = 1  − 1.67·3-s + 0.447·5-s + 0.341·7-s + 1.80·9-s − 1.66·11-s − 0.172·13-s − 0.749·15-s + 0.855·17-s − 0.251·19-s − 0.572·21-s + 1.11·23-s + 0.200·25-s − 1.35·27-s − 0.185·29-s − 0.297·31-s + 2.79·33-s + 0.152·35-s − 0.374·37-s + 0.289·39-s − 1.19·41-s − 0.167·43-s + 0.809·45-s − 0.241·47-s − 0.883·49-s − 1.43·51-s − 0.333·53-s − 0.745·55-s + ⋯

Functional equation

Λ(s)=(1160s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1160s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11601160    =    23⋅5⋅292^{3} \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 9.262649.26264
Root analytic conductor: 3.043453.04345
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1160, ( :1/2), −1)(2,\ 1160,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
5 1−T 1 - T
29 1+T 1 + T
good3 1+2.90T+3T2 1 + 2.90T + 3T^{2}
7 1−0.903T+7T2 1 - 0.903T + 7T^{2}
11 1+5.52T+11T2 1 + 5.52T + 11T^{2}
13 1+0.622T+13T2 1 + 0.622T + 13T^{2}
17 1−3.52T+17T2 1 - 3.52T + 17T^{2}
19 1+1.09T+19T2 1 + 1.09T + 19T^{2}
23 1−5.33T+23T2 1 - 5.33T + 23T^{2}
31 1+1.65T+31T2 1 + 1.65T + 31T^{2}
37 1+2.28T+37T2 1 + 2.28T + 37T^{2}
41 1+7.67T+41T2 1 + 7.67T + 41T^{2}
43 1+1.09T+43T2 1 + 1.09T + 43T^{2}
47 1+1.65T+47T2 1 + 1.65T + 47T^{2}
53 1+2.42T+53T2 1 + 2.42T + 53T^{2}
59 1−9.28T+59T2 1 - 9.28T + 59T^{2}
61 1+10.9T+61T2 1 + 10.9T + 61T^{2}
67 1+11.1T+67T2 1 + 11.1T + 67T^{2}
71 1+7.18T+71T2 1 + 7.18T + 71T^{2}
73 1−11.9T+73T2 1 - 11.9T + 73T^{2}
79 1+15.3T+79T2 1 + 15.3T + 79T^{2}
83 1+7.95T+83T2 1 + 7.95T + 83T^{2}
89 1+16.6T+89T2 1 + 16.6T + 89T^{2}
97 1+11.9T+97T2 1 + 11.9T + 97T^{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

−9.775592692298337126488356008762, −8.489508094732840610101553209066, −7.51629027743510684930393975044, −6.77864928260940692321226489053, −5.72132808141391004203115933649, −5.28200868374108976956855982509, −4.60275181891390001987840159542, −2.99132596284260544879359330678, −1.50414343902988283527027987869, 0, 1.50414343902988283527027987869, 2.99132596284260544879359330678, 4.60275181891390001987840159542, 5.28200868374108976956855982509, 5.72132808141391004203115933649, 6.77864928260940692321226489053, 7.51629027743510684930393975044, 8.489508094732840610101553209066, 9.775592692298337126488356008762

Graph of the ZZ-function along the critical line