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Properties

Label 2-1160-1.1-c3-0-70
Degree 22
Conductor 11601160
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.40·3-s − 5·5-s − 15.8·7-s + 2.26·9-s + 20.7·11-s + 40.3·13-s − 27.0·15-s + 40.6·17-s − 27.4·19-s − 85.6·21-s − 144.·23-s + 25·25-s − 133.·27-s − 29·29-s + 190.·31-s + 112.·33-s + 79.1·35-s + 65.8·37-s + 218.·39-s − 291.·41-s − 274.·43-s − 11.3·45-s + 32.1·47-s − 92.4·49-s + 219.·51-s + 407.·53-s − 103.·55-s + ⋯
L(s)  = 1  + 1.04·3-s − 0.447·5-s − 0.854·7-s + 0.0837·9-s + 0.567·11-s + 0.860·13-s − 0.465·15-s + 0.580·17-s − 0.330·19-s − 0.889·21-s − 1.30·23-s + 0.200·25-s − 0.953·27-s − 0.185·29-s + 1.10·31-s + 0.590·33-s + 0.382·35-s + 0.292·37-s + 0.895·39-s − 1.10·41-s − 0.974·43-s − 0.0374·45-s + 0.0999·47-s − 0.269·49-s + 0.603·51-s + 1.05·53-s − 0.253·55-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: −1-1
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: 11
Selberg data: (2, 1160, ( :3/2), −1)(2,\ 1160,\ (\ :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+5T 1 + 5T
29 1+29T 1 + 29T
good3 1−5.40T+27T2 1 - 5.40T + 27T^{2}
7 1+15.8T+343T2 1 + 15.8T + 343T^{2}
11 1−20.7T+1.33e3T2 1 - 20.7T + 1.33e3T^{2}
13 1−40.3T+2.19e3T2 1 - 40.3T + 2.19e3T^{2}
17 1−40.6T+4.91e3T2 1 - 40.6T + 4.91e3T^{2}
19 1+27.4T+6.85e3T2 1 + 27.4T + 6.85e3T^{2}
23 1+144.T+1.21e4T2 1 + 144.T + 1.21e4T^{2}
31 1−190.T+2.97e4T2 1 - 190.T + 2.97e4T^{2}
37 1−65.8T+5.06e4T2 1 - 65.8T + 5.06e4T^{2}
41 1+291.T+6.89e4T2 1 + 291.T + 6.89e4T^{2}
43 1+274.T+7.95e4T2 1 + 274.T + 7.95e4T^{2}
47 1−32.1T+1.03e5T2 1 - 32.1T + 1.03e5T^{2}
53 1−407.T+1.48e5T2 1 - 407.T + 1.48e5T^{2}
59 1−223.T+2.05e5T2 1 - 223.T + 2.05e5T^{2}
61 1+717.T+2.26e5T2 1 + 717.T + 2.26e5T^{2}
67 1+684.T+3.00e5T2 1 + 684.T + 3.00e5T^{2}
71 1+462.T+3.57e5T2 1 + 462.T + 3.57e5T^{2}
73 1+551.T+3.89e5T2 1 + 551.T + 3.89e5T^{2}
79 1+529.T+4.93e5T2 1 + 529.T + 4.93e5T^{2}
83 1+814.T+5.71e5T2 1 + 814.T + 5.71e5T^{2}
89 1+24.8T+7.04e5T2 1 + 24.8T + 7.04e5T^{2}
97 1+1.49e3T+9.12e5T2 1 + 1.49e3T + 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

−8.816436108257124544102847005812, −8.360867974740071791021955013109, −7.52341676533148195407732181506, −6.50518415325226059004726269437, −5.79866407409173403305738799128, −4.29773992653080921008112919019, −3.54428738920380317610729779475, −2.85530550695809413871618828389, −1.54090611018359539899695571423, 0, 1.54090611018359539899695571423, 2.85530550695809413871618828389, 3.54428738920380317610729779475, 4.29773992653080921008112919019, 5.79866407409173403305738799128, 6.50518415325226059004726269437, 7.52341676533148195407732181506, 8.360867974740071791021955013109, 8.816436108257124544102847005812

Graph of the ZZ-function along the critical line