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Properties

Label 2-1232-1.1-c3-0-68
Degree 22
Conductor 12321232
Sign −1-1
Analytic cond. 72.690372.6903
Root an. cond. 8.525868.52586
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.34·3-s + 3.97·5-s + 7·7-s − 25.2·9-s + 11·11-s + 26.5·13-s − 5.33·15-s − 90.7·17-s + 68.4·19-s − 9.39·21-s + 36.3·23-s − 109.·25-s + 70.0·27-s + 176.·29-s − 177.·31-s − 14.7·33-s + 27.8·35-s − 299.·37-s − 35.6·39-s − 45.3·41-s + 528.·43-s − 100.·45-s − 357.·47-s + 49·49-s + 121.·51-s − 742.·53-s + 43.7·55-s + ⋯
L(s)  = 1  − 0.258·3-s + 0.355·5-s + 0.377·7-s − 0.933·9-s + 0.301·11-s + 0.566·13-s − 0.0918·15-s − 1.29·17-s + 0.825·19-s − 0.0975·21-s + 0.329·23-s − 0.873·25-s + 0.499·27-s + 1.12·29-s − 1.02·31-s − 0.0778·33-s + 0.134·35-s − 1.32·37-s − 0.146·39-s − 0.172·41-s + 1.87·43-s − 0.331·45-s − 1.10·47-s + 0.142·49-s + 0.334·51-s − 1.92·53-s + 0.107·55-s + ⋯

Functional equation

Λ(s)=(1232s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1232s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12321232    =    24⋅7⋅112^{4} \cdot 7 \cdot 11
Sign: −1-1
Analytic conductor: 72.690372.6903
Root analytic conductor: 8.525868.52586
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1232, ( :3/2), −1)(2,\ 1232,\ (\ :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
7 1−7T 1 - 7T
11 1−11T 1 - 11T
good3 1+1.34T+27T2 1 + 1.34T + 27T^{2}
5 1−3.97T+125T2 1 - 3.97T + 125T^{2}
13 1−26.5T+2.19e3T2 1 - 26.5T + 2.19e3T^{2}
17 1+90.7T+4.91e3T2 1 + 90.7T + 4.91e3T^{2}
19 1−68.4T+6.85e3T2 1 - 68.4T + 6.85e3T^{2}
23 1−36.3T+1.21e4T2 1 - 36.3T + 1.21e4T^{2}
29 1−176.T+2.43e4T2 1 - 176.T + 2.43e4T^{2}
31 1+177.T+2.97e4T2 1 + 177.T + 2.97e4T^{2}
37 1+299.T+5.06e4T2 1 + 299.T + 5.06e4T^{2}
41 1+45.3T+6.89e4T2 1 + 45.3T + 6.89e4T^{2}
43 1−528.T+7.95e4T2 1 - 528.T + 7.95e4T^{2}
47 1+357.T+1.03e5T2 1 + 357.T + 1.03e5T^{2}
53 1+742.T+1.48e5T2 1 + 742.T + 1.48e5T^{2}
59 1−877.T+2.05e5T2 1 - 877.T + 2.05e5T^{2}
61 1−199.T+2.26e5T2 1 - 199.T + 2.26e5T^{2}
67 1+998.T+3.00e5T2 1 + 998.T + 3.00e5T^{2}
71 1−27.8T+3.57e5T2 1 - 27.8T + 3.57e5T^{2}
73 1+161.T+3.89e5T2 1 + 161.T + 3.89e5T^{2}
79 1−624.T+4.93e5T2 1 - 624.T + 4.93e5T^{2}
83 1+396.T+5.71e5T2 1 + 396.T + 5.71e5T^{2}
89 1+1.40e3T+7.04e5T2 1 + 1.40e3T + 7.04e5T^{2}
97 1+193.T+9.12e5T2 1 + 193.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

−8.886503544550114050148766949211, −8.293128098734129805926615470389, −7.20599644483045384026345668609, −6.33472297247191506004952394664, −5.60150641882922079421586460391, −4.75589468995490599786438338057, −3.63902757598236937799495658977, −2.53078046987111218618388731687, −1.39158271378551134143670489421, 0, 1.39158271378551134143670489421, 2.53078046987111218618388731687, 3.63902757598236937799495658977, 4.75589468995490599786438338057, 5.60150641882922079421586460391, 6.33472297247191506004952394664, 7.20599644483045384026345668609, 8.293128098734129805926615470389, 8.886503544550114050148766949211

Graph of the ZZ-function along the critical line