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Properties

Label 2-115-1.1-c1-0-5
Degree 22
Conductor 115115
Sign 11
Analytic cond. 0.9182790.918279
Root an. cond. 0.9582690.958269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.32·2-s + 1.56·3-s − 0.231·4-s + 5-s + 2.07·6-s − 3.50·7-s − 2.96·8-s − 0.561·9-s + 1.32·10-s − 0.659·11-s − 0.362·12-s + 5.91·13-s − 4.65·14-s + 1.56·15-s − 3.48·16-s + 0.844·17-s − 0.746·18-s + 0.659·19-s − 0.231·20-s − 5.47·21-s − 0.876·22-s − 23-s − 4.63·24-s + 25-s + 7.85·26-s − 5.56·27-s + 0.812·28-s + ⋯
L(s)  = 1  + 0.940·2-s + 0.901·3-s − 0.115·4-s + 0.447·5-s + 0.847·6-s − 1.32·7-s − 1.04·8-s − 0.187·9-s + 0.420·10-s − 0.198·11-s − 0.104·12-s + 1.63·13-s − 1.24·14-s + 0.403·15-s − 0.870·16-s + 0.204·17-s − 0.176·18-s + 0.151·19-s − 0.0518·20-s − 1.19·21-s − 0.186·22-s − 0.208·23-s − 0.945·24-s + 0.200·25-s + 1.54·26-s − 1.07·27-s + 0.153·28-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 11
Analytic conductor: 0.9182790.918279
Root analytic conductor: 0.9582690.958269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 115, ( :1/2), 1)(2,\ 115,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7122186921.712218692
L(12)L(\frac12) ≈\approx 1.7122186921.712218692
L(32)L(\frac{3}{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−T 1 - T
23 1+T 1 + T
good2 1−1.32T+2T2 1 - 1.32T + 2T^{2}
3 1−1.56T+3T2 1 - 1.56T + 3T^{2}
7 1+3.50T+7T2 1 + 3.50T + 7T^{2}
11 1+0.659T+11T2 1 + 0.659T + 11T^{2}
13 1−5.91T+13T2 1 - 5.91T + 13T^{2}
17 1−0.844T+17T2 1 - 0.844T + 17T^{2}
19 1−0.659T+19T2 1 - 0.659T + 19T^{2}
29 1−1.59T+29T2 1 - 1.59T + 29T^{2}
31 1−6.75T+31T2 1 - 6.75T + 31T^{2}
37 1+11.7T+37T2 1 + 11.7T + 37T^{2}
41 1−6.40T+41T2 1 - 6.40T + 41T^{2}
43 1+9.47T+43T2 1 + 9.47T + 43T^{2}
47 1−6.88T+47T2 1 - 6.88T + 47T^{2}
53 1−6.64T+53T2 1 - 6.64T + 53T^{2}
59 1+4.97T+59T2 1 + 4.97T + 59T^{2}
61 1−5.78T+61T2 1 - 5.78T + 61T^{2}
67 1−8.31T+67T2 1 - 8.31T + 67T^{2}
71 1+3.63T+71T2 1 + 3.63T + 71T^{2}
73 1+13.9T+73T2 1 + 13.9T + 73T^{2}
79 1−1.02T+79T2 1 - 1.02T + 79T^{2}
83 1+8.27T+83T2 1 + 8.27T + 83T^{2}
89 1−17.6T+89T2 1 - 17.6T + 89T^{2}
97 1+8.34T+97T2 1 + 8.34T + 97T^{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

−13.57963036176141021296510619764, −13.04634885074001251790285594339, −11.88144176656244784495546319528, −10.27472711198260801204894879963, −9.175151842257403588921190550651, −8.437168023621527615153667015029, −6.53762544700321598932036337105, −5.61870438256016155150094705080, −3.79699061686296646681914444342, −2.93276387909680137409825122262, 2.93276387909680137409825122262, 3.79699061686296646681914444342, 5.61870438256016155150094705080, 6.53762544700321598932036337105, 8.437168023621527615153667015029, 9.175151842257403588921190550651, 10.27472711198260801204894879963, 11.88144176656244784495546319528, 13.04634885074001251790285594339, 13.57963036176141021296510619764

Graph of the ZZ-function along the critical line