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Properties

Label 2-77e2-1.1-c1-0-100
Degree 22
Conductor 59295929
Sign 11
Analytic cond. 47.343347.3433
Root an. cond. 6.880646.88064
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  − 1.87·2-s + 0.652·3-s + 1.53·4-s + 3.53·5-s − 1.22·6-s + 0.879·8-s − 2.57·9-s − 6.63·10-s + 12-s − 4.41·13-s + 2.30·15-s − 4.71·16-s + 5.24·17-s + 4.83·18-s − 1.81·19-s + 5.41·20-s − 6.33·23-s + 0.573·24-s + 7.47·25-s + 8.29·26-s − 3.63·27-s − 1.92·29-s − 4.33·30-s + 1.46·31-s + 7.10·32-s − 9.86·34-s − 3.94·36-s + ⋯
L(s)  = 1  − 1.32·2-s + 0.376·3-s + 0.766·4-s + 1.57·5-s − 0.500·6-s + 0.310·8-s − 0.857·9-s − 2.09·10-s + 0.288·12-s − 1.22·13-s + 0.595·15-s − 1.17·16-s + 1.27·17-s + 1.14·18-s − 0.416·19-s + 1.21·20-s − 1.32·23-s + 0.117·24-s + 1.49·25-s + 1.62·26-s − 0.700·27-s − 0.356·29-s − 0.791·30-s + 0.263·31-s + 1.25·32-s − 1.69·34-s − 0.657·36-s + ⋯

Functional equation

Λ(s)=(5929s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5929s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 59295929    =    72⋅1127^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 47.343347.3433
Root analytic conductor: 6.880646.88064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5929, ( :1/2), 1)(2,\ 5929,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1953682211.195368221
L(12)L(\frac12) ≈\approx 1.1953682211.195368221
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)
bad7 1 1
11 1 1
good2 1+1.87T+2T2 1 + 1.87T + 2T^{2}
3 1−0.652T+3T2 1 - 0.652T + 3T^{2}
5 1−3.53T+5T2 1 - 3.53T + 5T^{2}
13 1+4.41T+13T2 1 + 4.41T + 13T^{2}
17 1−5.24T+17T2 1 - 5.24T + 17T^{2}
19 1+1.81T+19T2 1 + 1.81T + 19T^{2}
23 1+6.33T+23T2 1 + 6.33T + 23T^{2}
29 1+1.92T+29T2 1 + 1.92T + 29T^{2}
31 1−1.46T+31T2 1 - 1.46T + 31T^{2}
37 1−4.45T+37T2 1 - 4.45T + 37T^{2}
41 1+0.283T+41T2 1 + 0.283T + 41T^{2}
43 1−3.41T+43T2 1 - 3.41T + 43T^{2}
47 1+4.55T+47T2 1 + 4.55T + 47T^{2}
53 1+7.23T+53T2 1 + 7.23T + 53T^{2}
59 1−9.53T+59T2 1 - 9.53T + 59T^{2}
61 1−1.14T+61T2 1 - 1.14T + 61T^{2}
67 1+0.694T+67T2 1 + 0.694T + 67T^{2}
71 1−9.46T+71T2 1 - 9.46T + 71T^{2}
73 1+2.34T+73T2 1 + 2.34T + 73T^{2}
79 1−12.4T+79T2 1 - 12.4T + 79T^{2}
83 1−11.3T+83T2 1 - 11.3T + 83T^{2}
89 1+3.46T+89T2 1 + 3.46T + 89T^{2}
97 1−15.3T+97T2 1 - 15.3T + 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

−8.079835112255427124009097962817, −7.77356511780602651213909592745, −6.79943295253231906256909164071, −6.03037587997585309496377109195, −5.43895345034856127105266532632, −4.59941758818199410289292186516, −3.29816500528366063744911202911, −2.26517481599899415457560392687, −1.95838985996695562947404658157, −0.67674583322847028935246894540, 0.67674583322847028935246894540, 1.95838985996695562947404658157, 2.26517481599899415457560392687, 3.29816500528366063744911202911, 4.59941758818199410289292186516, 5.43895345034856127105266532632, 6.03037587997585309496377109195, 6.79943295253231906256909164071, 7.77356511780602651213909592745, 8.079835112255427124009097962817

Graph of the ZZ-function along the critical line