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Properties

Label 2-1232-1.1-c3-0-58
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  − 10.1·3-s + 8.69·5-s + 7·7-s + 75.9·9-s + 11·11-s − 76.3·13-s − 88.2·15-s + 39.7·17-s + 27.9·19-s − 71.0·21-s − 87.2·23-s − 49.3·25-s − 496.·27-s − 38.3·29-s + 186.·31-s − 111.·33-s + 60.8·35-s − 218.·37-s + 774.·39-s + 80.1·41-s + 35.1·43-s + 660.·45-s + 282.·47-s + 49·49-s − 403.·51-s + 145.·53-s + 95.6·55-s + ⋯
L(s)  = 1  − 1.95·3-s + 0.778·5-s + 0.377·7-s + 2.81·9-s + 0.301·11-s − 1.62·13-s − 1.51·15-s + 0.566·17-s + 0.337·19-s − 0.738·21-s − 0.790·23-s − 0.394·25-s − 3.53·27-s − 0.245·29-s + 1.07·31-s − 0.588·33-s + 0.294·35-s − 0.972·37-s + 3.18·39-s + 0.305·41-s + 0.124·43-s + 2.18·45-s + 0.877·47-s + 0.142·49-s − 1.10·51-s + 0.376·53-s + 0.234·55-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+10.1T+27T2 1 + 10.1T + 27T^{2}
5 1−8.69T+125T2 1 - 8.69T + 125T^{2}
13 1+76.3T+2.19e3T2 1 + 76.3T + 2.19e3T^{2}
17 1−39.7T+4.91e3T2 1 - 39.7T + 4.91e3T^{2}
19 1−27.9T+6.85e3T2 1 - 27.9T + 6.85e3T^{2}
23 1+87.2T+1.21e4T2 1 + 87.2T + 1.21e4T^{2}
29 1+38.3T+2.43e4T2 1 + 38.3T + 2.43e4T^{2}
31 1−186.T+2.97e4T2 1 - 186.T + 2.97e4T^{2}
37 1+218.T+5.06e4T2 1 + 218.T + 5.06e4T^{2}
41 1−80.1T+6.89e4T2 1 - 80.1T + 6.89e4T^{2}
43 1−35.1T+7.95e4T2 1 - 35.1T + 7.95e4T^{2}
47 1−282.T+1.03e5T2 1 - 282.T + 1.03e5T^{2}
53 1−145.T+1.48e5T2 1 - 145.T + 1.48e5T^{2}
59 1+91.0T+2.05e5T2 1 + 91.0T + 2.05e5T^{2}
61 1−808.T+2.26e5T2 1 - 808.T + 2.26e5T^{2}
67 1+794.T+3.00e5T2 1 + 794.T + 3.00e5T^{2}
71 1+946.T+3.57e5T2 1 + 946.T + 3.57e5T^{2}
73 1−801.T+3.89e5T2 1 - 801.T + 3.89e5T^{2}
79 1−890.T+4.93e5T2 1 - 890.T + 4.93e5T^{2}
83 1−559.T+5.71e5T2 1 - 559.T + 5.71e5T^{2}
89 1+1.52e3T+7.04e5T2 1 + 1.52e3T + 7.04e5T^{2}
97 1−664.T+9.12e5T2 1 - 664.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

−9.331365291576990630597634661955, −7.78520653904640543621688212872, −7.08985415769147355126255954123, −6.24664216259771538233751562328, −5.51448642916635854182502317376, −4.97986559769831821620795053815, −4.06157890309676685964490331456, −2.22255664627633354396678006536, −1.16552832259182026736239862710, 0, 1.16552832259182026736239862710, 2.22255664627633354396678006536, 4.06157890309676685964490331456, 4.97986559769831821620795053815, 5.51448642916635854182502317376, 6.24664216259771538233751562328, 7.08985415769147355126255954123, 7.78520653904640543621688212872, 9.331365291576990630597634661955

Graph of the ZZ-function along the critical line