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Properties

Label 2-1160-1.1-c1-0-8
Degree 22
Conductor 11601160
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.37·3-s + 5-s + 4.57·7-s + 2.66·9-s + 5.84·11-s + 2.86·13-s − 2.37·15-s − 6.90·17-s − 2.24·19-s − 10.8·21-s − 0.981·23-s + 25-s + 0.798·27-s + 29-s + 4.18·31-s − 13.9·33-s + 4.57·35-s + 2.31·37-s − 6.81·39-s − 7.15·41-s − 4.90·43-s + 2.66·45-s + 6.85·47-s + 13.9·49-s + 16.4·51-s − 6.06·53-s + 5.84·55-s + ⋯
L(s)  = 1  − 1.37·3-s + 0.447·5-s + 1.73·7-s + 0.888·9-s + 1.76·11-s + 0.794·13-s − 0.614·15-s − 1.67·17-s − 0.514·19-s − 2.37·21-s − 0.204·23-s + 0.200·25-s + 0.153·27-s + 0.185·29-s + 0.751·31-s − 2.42·33-s + 0.773·35-s + 0.380·37-s − 1.09·39-s − 1.11·41-s − 0.748·43-s + 0.397·45-s + 0.999·47-s + 1.99·49-s + 2.30·51-s − 0.833·53-s + 0.788·55-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: 11
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: 00
Selberg data: (2, 1160, ( :1/2), 1)(2,\ 1160,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4772728151.477272815
L(12)L(\frac12) ≈\approx 1.4772728151.477272815
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+2.37T+3T2 1 + 2.37T + 3T^{2}
7 1−4.57T+7T2 1 - 4.57T + 7T^{2}
11 1−5.84T+11T2 1 - 5.84T + 11T^{2}
13 1−2.86T+13T2 1 - 2.86T + 13T^{2}
17 1+6.90T+17T2 1 + 6.90T + 17T^{2}
19 1+2.24T+19T2 1 + 2.24T + 19T^{2}
23 1+0.981T+23T2 1 + 0.981T + 23T^{2}
31 1−4.18T+31T2 1 - 4.18T + 31T^{2}
37 1−2.31T+37T2 1 - 2.31T + 37T^{2}
41 1+7.15T+41T2 1 + 7.15T + 41T^{2}
43 1+4.90T+43T2 1 + 4.90T + 43T^{2}
47 1−6.85T+47T2 1 - 6.85T + 47T^{2}
53 1+6.06T+53T2 1 + 6.06T + 53T^{2}
59 1−10.8T+59T2 1 - 10.8T + 59T^{2}
61 1−10.9T+61T2 1 - 10.9T + 61T^{2}
67 1−7.30T+67T2 1 - 7.30T + 67T^{2}
71 1−9.51T+71T2 1 - 9.51T + 71T^{2}
73 1+12.7T+73T2 1 + 12.7T + 73T^{2}
79 1−2.44T+79T2 1 - 2.44T + 79T^{2}
83 1+9.93T+83T2 1 + 9.93T + 83T^{2}
89 1+9.41T+89T2 1 + 9.41T + 89T^{2}
97 1+4.95T+97T2 1 + 4.95T + 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

−9.990738479409832739612090624503, −8.770056319760676477320354069320, −8.413592644719380525798106885673, −6.88900402700430252443189005328, −6.45416110907705669585843371809, −5.57226137701354809297742460387, −4.66250519650252249717173174536, −4.08062410984427401847682360068, −2.00443879036301404979573102681, −1.08294990422208240121632186307, 1.08294990422208240121632186307, 2.00443879036301404979573102681, 4.08062410984427401847682360068, 4.66250519650252249717173174536, 5.57226137701354809297742460387, 6.45416110907705669585843371809, 6.88900402700430252443189005328, 8.413592644719380525798106885673, 8.770056319760676477320354069320, 9.990738479409832739612090624503

Graph of the ZZ-function along the critical line