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Properties

Label 2-1160-1.1-c3-0-2
Degree 22
Conductor 11601160
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.528·3-s − 5·5-s − 17.5·7-s − 26.7·9-s − 24.1·11-s − 79.1·13-s + 2.64·15-s − 6.40·17-s − 127.·19-s + 9.26·21-s − 52.0·23-s + 25·25-s + 28.3·27-s + 29·29-s + 89.9·31-s + 12.7·33-s + 87.6·35-s − 71.2·37-s + 41.8·39-s + 34.0·41-s + 355.·43-s + 133.·45-s − 600.·47-s − 35.7·49-s + 3.38·51-s + 738.·53-s + 120.·55-s + ⋯
L(s)  = 1  − 0.101·3-s − 0.447·5-s − 0.946·7-s − 0.989·9-s − 0.662·11-s − 1.68·13-s + 0.0454·15-s − 0.0913·17-s − 1.54·19-s + 0.0962·21-s − 0.472·23-s + 0.200·25-s + 0.202·27-s + 0.185·29-s + 0.521·31-s + 0.0673·33-s + 0.423·35-s − 0.316·37-s + 0.171·39-s + 0.129·41-s + 1.26·43-s + 0.442·45-s − 1.86·47-s − 0.104·49-s + 0.00929·51-s + 1.91·53-s + 0.296·55-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: 11
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: 00
Selberg data: (2, 1160, ( :3/2), 1)(2,\ 1160,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.23472526540.2347252654
L(12)L(\frac12) ≈\approx 0.23472526540.2347252654
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+0.528T+27T2 1 + 0.528T + 27T^{2}
7 1+17.5T+343T2 1 + 17.5T + 343T^{2}
11 1+24.1T+1.33e3T2 1 + 24.1T + 1.33e3T^{2}
13 1+79.1T+2.19e3T2 1 + 79.1T + 2.19e3T^{2}
17 1+6.40T+4.91e3T2 1 + 6.40T + 4.91e3T^{2}
19 1+127.T+6.85e3T2 1 + 127.T + 6.85e3T^{2}
23 1+52.0T+1.21e4T2 1 + 52.0T + 1.21e4T^{2}
31 1−89.9T+2.97e4T2 1 - 89.9T + 2.97e4T^{2}
37 1+71.2T+5.06e4T2 1 + 71.2T + 5.06e4T^{2}
41 1−34.0T+6.89e4T2 1 - 34.0T + 6.89e4T^{2}
43 1−355.T+7.95e4T2 1 - 355.T + 7.95e4T^{2}
47 1+600.T+1.03e5T2 1 + 600.T + 1.03e5T^{2}
53 1−738.T+1.48e5T2 1 - 738.T + 1.48e5T^{2}
59 1+151.T+2.05e5T2 1 + 151.T + 2.05e5T^{2}
61 1+581.T+2.26e5T2 1 + 581.T + 2.26e5T^{2}
67 1−74.5T+3.00e5T2 1 - 74.5T + 3.00e5T^{2}
71 1−281.T+3.57e5T2 1 - 281.T + 3.57e5T^{2}
73 1+578.T+3.89e5T2 1 + 578.T + 3.89e5T^{2}
79 1−549.T+4.93e5T2 1 - 549.T + 4.93e5T^{2}
83 1+76.0T+5.71e5T2 1 + 76.0T + 5.71e5T^{2}
89 1+406.T+7.04e5T2 1 + 406.T + 7.04e5T^{2}
97 1+1.16e3T+9.12e5T2 1 + 1.16e3T + 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

−9.463269640967706047176590201820, −8.541211019965665049133479188100, −7.81993325378592267584047433273, −6.88293269621421063884655919591, −6.10639413846455198459391047531, −5.13695553550460893813683844552, −4.22153237544055161836660713019, −2.99829569636976870056863295574, −2.32630673924041099380254854433, −0.22989929427295113686550148219, 0.22989929427295113686550148219, 2.32630673924041099380254854433, 2.99829569636976870056863295574, 4.22153237544055161836660713019, 5.13695553550460893813683844552, 6.10639413846455198459391047531, 6.88293269621421063884655919591, 7.81993325378592267584047433273, 8.541211019965665049133479188100, 9.463269640967706047176590201820

Graph of the ZZ-function along the critical line