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Properties

Label 2-1232-1.1-c3-0-65
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  + 2·3-s − 12.1·5-s + 7·7-s − 23·9-s − 11·11-s + 32.3·13-s − 24.3·15-s − 42.4·17-s + 127.·19-s + 14·21-s + 202.·23-s + 22.9·25-s − 100·27-s + 126.·29-s + 4.16·31-s − 22·33-s − 85.1·35-s − 111.·37-s + 64.6·39-s + 233.·41-s − 476.·43-s + 279.·45-s − 550.·47-s + 49·49-s − 84.9·51-s − 138.·53-s + 133.·55-s + ⋯
L(s)  = 1  + 0.384·3-s − 1.08·5-s + 0.377·7-s − 0.851·9-s − 0.301·11-s + 0.689·13-s − 0.418·15-s − 0.606·17-s + 1.53·19-s + 0.145·21-s + 1.83·23-s + 0.183·25-s − 0.712·27-s + 0.813·29-s + 0.0241·31-s − 0.116·33-s − 0.411·35-s − 0.494·37-s + 0.265·39-s + 0.889·41-s − 1.69·43-s + 0.926·45-s − 1.70·47-s + 0.142·49-s − 0.233·51-s − 0.360·53-s + 0.328·55-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 1−2T+27T2 1 - 2T + 27T^{2}
5 1+12.1T+125T2 1 + 12.1T + 125T^{2}
13 1−32.3T+2.19e3T2 1 - 32.3T + 2.19e3T^{2}
17 1+42.4T+4.91e3T2 1 + 42.4T + 4.91e3T^{2}
19 1−127.T+6.85e3T2 1 - 127.T + 6.85e3T^{2}
23 1−202.T+1.21e4T2 1 - 202.T + 1.21e4T^{2}
29 1−126.T+2.43e4T2 1 - 126.T + 2.43e4T^{2}
31 1−4.16T+2.97e4T2 1 - 4.16T + 2.97e4T^{2}
37 1+111.T+5.06e4T2 1 + 111.T + 5.06e4T^{2}
41 1−233.T+6.89e4T2 1 - 233.T + 6.89e4T^{2}
43 1+476.T+7.95e4T2 1 + 476.T + 7.95e4T^{2}
47 1+550.T+1.03e5T2 1 + 550.T + 1.03e5T^{2}
53 1+138.T+1.48e5T2 1 + 138.T + 1.48e5T^{2}
59 1+276.T+2.05e5T2 1 + 276.T + 2.05e5T^{2}
61 1+619.T+2.26e5T2 1 + 619.T + 2.26e5T^{2}
67 1−127.T+3.00e5T2 1 - 127.T + 3.00e5T^{2}
71 1−464.T+3.57e5T2 1 - 464.T + 3.57e5T^{2}
73 1−103.T+3.89e5T2 1 - 103.T + 3.89e5T^{2}
79 1+1.20e3T+4.93e5T2 1 + 1.20e3T + 4.93e5T^{2}
83 1+1.24e3T+5.71e5T2 1 + 1.24e3T + 5.71e5T^{2}
89 1−931.T+7.04e5T2 1 - 931.T + 7.04e5T^{2}
97 1+1.03e3T+9.12e5T2 1 + 1.03e3T + 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

−8.738845106108238624611445375660, −8.181999593543837523931289843218, −7.47796766157947515697691584644, −6.57987725302771674719848964950, −5.38828003209066394005868420055, −4.63260282289843914274474593876, −3.43698223170941542948787903473, −2.89524194664079758459752015036, −1.30494788038398828130851024396, 0, 1.30494788038398828130851024396, 2.89524194664079758459752015036, 3.43698223170941542948787903473, 4.63260282289843914274474593876, 5.38828003209066394005868420055, 6.57987725302771674719848964950, 7.47796766157947515697691584644, 8.181999593543837523931289843218, 8.738845106108238624611445375660

Graph of the ZZ-function along the critical line