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Properties

Label 2-115-1.1-c3-0-0
Degree 22
Conductor 115115
Sign 11
Analytic cond. 6.785216.78521
Root an. cond. 2.604842.60484
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  − 2.93·2-s − 3.85·3-s + 0.637·4-s − 5·5-s + 11.3·6-s − 23.5·7-s + 21.6·8-s − 12.1·9-s + 14.6·10-s − 58.2·11-s − 2.45·12-s + 68.5·13-s + 69.3·14-s + 19.2·15-s − 68.6·16-s + 101.·17-s + 35.6·18-s − 7.02·19-s − 3.18·20-s + 91.0·21-s + 171.·22-s − 23·23-s − 83.4·24-s + 25·25-s − 201.·26-s + 150.·27-s − 15.0·28-s + ⋯
L(s)  = 1  − 1.03·2-s − 0.742·3-s + 0.0797·4-s − 0.447·5-s + 0.771·6-s − 1.27·7-s + 0.956·8-s − 0.448·9-s + 0.464·10-s − 1.59·11-s − 0.0591·12-s + 1.46·13-s + 1.32·14-s + 0.332·15-s − 1.07·16-s + 1.44·17-s + 0.466·18-s − 0.0848·19-s − 0.0356·20-s + 0.945·21-s + 1.65·22-s − 0.208·23-s − 0.709·24-s + 0.200·25-s − 1.51·26-s + 1.07·27-s − 0.101·28-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 11
Analytic conductor: 6.785216.78521
Root analytic conductor: 2.604842.60484
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 115, ( :3/2), 1)(2,\ 115,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.33933700620.3393370062
L(12)L(\frac12) ≈\approx 0.33933700620.3393370062
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)
bad5 1+5T 1 + 5T
23 1+23T 1 + 23T
good2 1+2.93T+8T2 1 + 2.93T + 8T^{2}
3 1+3.85T+27T2 1 + 3.85T + 27T^{2}
7 1+23.5T+343T2 1 + 23.5T + 343T^{2}
11 1+58.2T+1.33e3T2 1 + 58.2T + 1.33e3T^{2}
13 1−68.5T+2.19e3T2 1 - 68.5T + 2.19e3T^{2}
17 1−101.T+4.91e3T2 1 - 101.T + 4.91e3T^{2}
19 1+7.02T+6.85e3T2 1 + 7.02T + 6.85e3T^{2}
29 1−206.T+2.43e4T2 1 - 206.T + 2.43e4T^{2}
31 1−54.8T+2.97e4T2 1 - 54.8T + 2.97e4T^{2}
37 1+241.T+5.06e4T2 1 + 241.T + 5.06e4T^{2}
41 1+122.T+6.89e4T2 1 + 122.T + 6.89e4T^{2}
43 1+320.T+7.95e4T2 1 + 320.T + 7.95e4T^{2}
47 1−107.T+1.03e5T2 1 - 107.T + 1.03e5T^{2}
53 1−127.T+1.48e5T2 1 - 127.T + 1.48e5T^{2}
59 1−693.T+2.05e5T2 1 - 693.T + 2.05e5T^{2}
61 1+899.T+2.26e5T2 1 + 899.T + 2.26e5T^{2}
67 1−110.T+3.00e5T2 1 - 110.T + 3.00e5T^{2}
71 1−225.T+3.57e5T2 1 - 225.T + 3.57e5T^{2}
73 1−746.T+3.89e5T2 1 - 746.T + 3.89e5T^{2}
79 1+1.09e3T+4.93e5T2 1 + 1.09e3T + 4.93e5T^{2}
83 1−1.28e3T+5.71e5T2 1 - 1.28e3T + 5.71e5T^{2}
89 1−1.20e3T+7.04e5T2 1 - 1.20e3T + 7.04e5T^{2}
97 1−903.T+9.12e5T2 1 - 903.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

−12.99298136631232983892396359156, −11.88669253397764018252067523996, −10.57890989989705518926454502166, −10.16209450217894625077240365764, −8.715476569558740201865894845728, −7.890627998094180743156842988206, −6.45859579729192296268086441367, −5.22482506741171296884994348225, −3.29939336007389935685831012019, −0.59393817036756763483328392026, 0.59393817036756763483328392026, 3.29939336007389935685831012019, 5.22482506741171296884994348225, 6.45859579729192296268086441367, 7.890627998094180743156842988206, 8.715476569558740201865894845728, 10.16209450217894625077240365764, 10.57890989989705518926454502166, 11.88669253397764018252067523996, 12.99298136631232983892396359156

Graph of the ZZ-function along the critical line