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Properties

Label 2-1160-1.1-c3-0-48
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  − 5.20·3-s − 5·5-s + 15.1·7-s + 0.117·9-s − 3.79·11-s + 21.8·13-s + 26.0·15-s + 5.42·17-s − 129.·19-s − 79.1·21-s + 58.3·23-s + 25·25-s + 139.·27-s + 29·29-s − 36.2·31-s + 19.7·33-s − 75.9·35-s + 182.·37-s − 113.·39-s − 189.·41-s + 238.·43-s − 0.586·45-s + 94.4·47-s − 112.·49-s − 28.2·51-s + 123.·53-s + 18.9·55-s + ⋯
L(s)  = 1  − 1.00·3-s − 0.447·5-s + 0.820·7-s + 0.00434·9-s − 0.104·11-s + 0.466·13-s + 0.448·15-s + 0.0773·17-s − 1.56·19-s − 0.822·21-s + 0.528·23-s + 0.200·25-s + 0.997·27-s + 0.185·29-s − 0.209·31-s + 0.104·33-s − 0.366·35-s + 0.811·37-s − 0.467·39-s − 0.720·41-s + 0.846·43-s − 0.00194·45-s + 0.292·47-s − 0.327·49-s − 0.0775·51-s + 0.319·53-s + 0.0465·55-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+5.20T+27T2 1 + 5.20T + 27T^{2}
7 1−15.1T+343T2 1 - 15.1T + 343T^{2}
11 1+3.79T+1.33e3T2 1 + 3.79T + 1.33e3T^{2}
13 1−21.8T+2.19e3T2 1 - 21.8T + 2.19e3T^{2}
17 1−5.42T+4.91e3T2 1 - 5.42T + 4.91e3T^{2}
19 1+129.T+6.85e3T2 1 + 129.T + 6.85e3T^{2}
23 1−58.3T+1.21e4T2 1 - 58.3T + 1.21e4T^{2}
31 1+36.2T+2.97e4T2 1 + 36.2T + 2.97e4T^{2}
37 1−182.T+5.06e4T2 1 - 182.T + 5.06e4T^{2}
41 1+189.T+6.89e4T2 1 + 189.T + 6.89e4T^{2}
43 1−238.T+7.95e4T2 1 - 238.T + 7.95e4T^{2}
47 1−94.4T+1.03e5T2 1 - 94.4T + 1.03e5T^{2}
53 1−123.T+1.48e5T2 1 - 123.T + 1.48e5T^{2}
59 1−836.T+2.05e5T2 1 - 836.T + 2.05e5T^{2}
61 1−667.T+2.26e5T2 1 - 667.T + 2.26e5T^{2}
67 1+657.T+3.00e5T2 1 + 657.T + 3.00e5T^{2}
71 1+313.T+3.57e5T2 1 + 313.T + 3.57e5T^{2}
73 1+573.T+3.89e5T2 1 + 573.T + 3.89e5T^{2}
79 1−408.T+4.93e5T2 1 - 408.T + 4.93e5T^{2}
83 1+273.T+5.71e5T2 1 + 273.T + 5.71e5T^{2}
89 1+1.44e3T+7.04e5T2 1 + 1.44e3T + 7.04e5T^{2}
97 1+73.6T+9.12e5T2 1 + 73.6T + 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

−8.740592605014439583000210438236, −8.321273210544961643182324042374, −7.25263369626383279386729731299, −6.39637843638587056975740482690, −5.58720537380202361384213691791, −4.75303622966634877976289811765, −3.94586692933649404916740673098, −2.52816332916665470416455967233, −1.16186218263449860805157123776, 0, 1.16186218263449860805157123776, 2.52816332916665470416455967233, 3.94586692933649404916740673098, 4.75303622966634877976289811765, 5.58720537380202361384213691791, 6.39637843638587056975740482690, 7.25263369626383279386729731299, 8.321273210544961643182324042374, 8.740592605014439583000210438236

Graph of the ZZ-function along the critical line