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Properties

Label 2-2320-1.1-c3-0-156
Degree 22
Conductor 23202320
Sign −1-1
Analytic cond. 136.884136.884
Root an. cond. 11.699711.6997
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s − 5·5-s + 14·7-s + 37·9-s − 62·11-s + 42·13-s − 40·15-s − 114·17-s + 70·19-s + 112·21-s − 62·23-s + 25·25-s + 80·27-s − 29·29-s − 142·31-s − 496·33-s − 70·35-s + 146·37-s + 336·39-s + 162·41-s − 352·43-s − 185·45-s + 444·47-s − 147·49-s − 912·51-s − 238·53-s + 310·55-s + ⋯
L(s)  = 1  + 1.53·3-s − 0.447·5-s + 0.755·7-s + 1.37·9-s − 1.69·11-s + 0.896·13-s − 0.688·15-s − 1.62·17-s + 0.845·19-s + 1.16·21-s − 0.562·23-s + 1/5·25-s + 0.570·27-s − 0.185·29-s − 0.822·31-s − 2.61·33-s − 0.338·35-s + 0.648·37-s + 1.37·39-s + 0.617·41-s − 1.24·43-s − 0.612·45-s + 1.37·47-s − 3/7·49-s − 2.50·51-s − 0.616·53-s + 0.760·55-s + ⋯

Functional equation

Λ(s)=(2320s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2320s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23202320    =    24⋅5⋅292^{4} \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 136.884136.884
Root analytic conductor: 11.699711.6997
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2320, ( :3/2), −1)(2,\ 2320,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+pT 1 + p T
29 1+pT 1 + p T
good3 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
7 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
11 1+62T+p3T2 1 + 62 T + p^{3} T^{2}
13 1−42T+p3T2 1 - 42 T + p^{3} T^{2}
17 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
19 1−70T+p3T2 1 - 70 T + p^{3} T^{2}
23 1+62T+p3T2 1 + 62 T + p^{3} T^{2}
31 1+142T+p3T2 1 + 142 T + p^{3} T^{2}
37 1−146T+p3T2 1 - 146 T + p^{3} T^{2}
41 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
43 1+352T+p3T2 1 + 352 T + p^{3} T^{2}
47 1−444T+p3T2 1 - 444 T + p^{3} T^{2}
53 1+238T+p3T2 1 + 238 T + p^{3} T^{2}
59 1+840T+p3T2 1 + 840 T + p^{3} T^{2}
61 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
67 1−154T+p3T2 1 - 154 T + p^{3} T^{2}
71 1+892T+p3T2 1 + 892 T + p^{3} T^{2}
73 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
79 1+1050T+p3T2 1 + 1050 T + p^{3} T^{2}
83 1−778T+p3T2 1 - 778 T + p^{3} T^{2}
89 1−1410T+p3T2 1 - 1410 T + p^{3} T^{2}
97 1−466T+p3T2 1 - 466 T + p^{3} T^{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

−8.114931425690514740569573413149, −7.85343645471940885447680708875, −7.09176589333032254614054280151, −5.86827057516832694327068891082, −4.84423863659621848699717762025, −4.10389421118643447789867394883, −3.17051820039886915576915139737, −2.43362398775419067404814876059, −1.57558170347222304912698906150, 0, 1.57558170347222304912698906150, 2.43362398775419067404814876059, 3.17051820039886915576915139737, 4.10389421118643447789867394883, 4.84423863659621848699717762025, 5.86827057516832694327068891082, 7.09176589333032254614054280151, 7.85343645471940885447680708875, 8.114931425690514740569573413149

Graph of the ZZ-function along the critical line