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Properties

Label 2-1160-1.1-c3-0-0
Degree 22
Conductor 11601160
Sign 11
Analytic cond. 68.442268.4422
Root an. cond. 8.272988.27298
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  − 9.64·3-s + 5·5-s − 34.7·7-s + 65.9·9-s − 12.5·11-s − 25.0·13-s − 48.2·15-s − 59.0·17-s − 61.8·19-s + 334.·21-s + 21.8·23-s + 25·25-s − 375.·27-s + 29·29-s − 319.·31-s + 121.·33-s − 173.·35-s − 370.·37-s + 241.·39-s − 490.·41-s + 213.·43-s + 329.·45-s − 408.·47-s + 862.·49-s + 568.·51-s + 67.8·53-s − 62.8·55-s + ⋯
L(s)  = 1  − 1.85·3-s + 0.447·5-s − 1.87·7-s + 2.44·9-s − 0.344·11-s − 0.534·13-s − 0.829·15-s − 0.841·17-s − 0.746·19-s + 3.47·21-s + 0.198·23-s + 0.200·25-s − 2.67·27-s + 0.185·29-s − 1.85·31-s + 0.638·33-s − 0.838·35-s − 1.64·37-s + 0.991·39-s − 1.86·41-s + 0.758·43-s + 1.09·45-s − 1.26·47-s + 2.51·49-s + 1.56·51-s + 0.175·53-s − 0.154·55-s + ⋯

Functional equation

Λ(s)=(1160s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1160s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 68.442268.4422
Root analytic conductor: 8.272988.27298
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1160, ( :3/2), 1)(2,\ 1160,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.0037419535580.003741953558
L(12)L(\frac12) ≈\approx 0.0037419535580.003741953558
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
5 1−5T 1 - 5T
29 1−29T 1 - 29T
good3 1+9.64T+27T2 1 + 9.64T + 27T^{2}
7 1+34.7T+343T2 1 + 34.7T + 343T^{2}
11 1+12.5T+1.33e3T2 1 + 12.5T + 1.33e3T^{2}
13 1+25.0T+2.19e3T2 1 + 25.0T + 2.19e3T^{2}
17 1+59.0T+4.91e3T2 1 + 59.0T + 4.91e3T^{2}
19 1+61.8T+6.85e3T2 1 + 61.8T + 6.85e3T^{2}
23 1−21.8T+1.21e4T2 1 - 21.8T + 1.21e4T^{2}
31 1+319.T+2.97e4T2 1 + 319.T + 2.97e4T^{2}
37 1+370.T+5.06e4T2 1 + 370.T + 5.06e4T^{2}
41 1+490.T+6.89e4T2 1 + 490.T + 6.89e4T^{2}
43 1−213.T+7.95e4T2 1 - 213.T + 7.95e4T^{2}
47 1+408.T+1.03e5T2 1 + 408.T + 1.03e5T^{2}
53 1−67.8T+1.48e5T2 1 - 67.8T + 1.48e5T^{2}
59 1+200.T+2.05e5T2 1 + 200.T + 2.05e5T^{2}
61 1+559.T+2.26e5T2 1 + 559.T + 2.26e5T^{2}
67 1+936.T+3.00e5T2 1 + 936.T + 3.00e5T^{2}
71 1−577.T+3.57e5T2 1 - 577.T + 3.57e5T^{2}
73 1+1.04e3T+3.89e5T2 1 + 1.04e3T + 3.89e5T^{2}
79 1+236.T+4.93e5T2 1 + 236.T + 4.93e5T^{2}
83 1+1.21e3T+5.71e5T2 1 + 1.21e3T + 5.71e5T^{2}
89 1−1.12e3T+7.04e5T2 1 - 1.12e3T + 7.04e5T^{2}
97 1+498.T+9.12e5T2 1 + 498.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

−9.670145429013768004668967709556, −8.900177438666038343083109806465, −7.17873629831697476223657439060, −6.76940155117212866247407078294, −6.05140371354414034524795934109, −5.38427966531705169744734702417, −4.44376163196853705610256767699, −3.26094953765759801370366883538, −1.79012664824546350147591471344, −0.03322278175610676689390214789, 0.03322278175610676689390214789, 1.79012664824546350147591471344, 3.26094953765759801370366883538, 4.44376163196853705610256767699, 5.38427966531705169744734702417, 6.05140371354414034524795934109, 6.76940155117212866247407078294, 7.17873629831697476223657439060, 8.900177438666038343083109806465, 9.670145429013768004668967709556

Graph of the ZZ-function along the critical line