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Properties

Label 2-85e2-1.1-c1-0-118
Degree 22
Conductor 72257225
Sign 11
Analytic cond. 57.691957.6919
Root an. cond. 7.595517.59551
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  + 0.704·2-s − 3.11·3-s − 1.50·4-s − 2.19·6-s + 4.06·7-s − 2.46·8-s + 6.73·9-s − 0.231·11-s + 4.69·12-s − 2.35·13-s + 2.86·14-s + 1.27·16-s + 4.74·18-s + 7.38·19-s − 12.6·21-s − 0.162·22-s + 1.87·23-s + 7.69·24-s − 1.65·26-s − 11.6·27-s − 6.11·28-s + 4.04·29-s + 0.130·31-s + 5.82·32-s + 0.720·33-s − 10.1·36-s + 5.82·37-s + ⋯
L(s)  = 1  + 0.497·2-s − 1.80·3-s − 0.752·4-s − 0.896·6-s + 1.53·7-s − 0.872·8-s + 2.24·9-s − 0.0696·11-s + 1.35·12-s − 0.653·13-s + 0.764·14-s + 0.317·16-s + 1.11·18-s + 1.69·19-s − 2.76·21-s − 0.0346·22-s + 0.391·23-s + 1.57·24-s − 0.325·26-s − 2.24·27-s − 1.15·28-s + 0.751·29-s + 0.0234·31-s + 1.03·32-s + 0.125·33-s − 1.68·36-s + 0.957·37-s + ⋯

Functional equation

Λ(s)=(7225s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7225s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 72257225    =    52⋅1725^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 57.691957.6919
Root analytic conductor: 7.595517.59551
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7225, ( :1/2), 1)(2,\ 7225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3536604051.353660405
L(12)L(\frac12) ≈\approx 1.3536604051.353660405
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 1
17 1 1
good2 1−0.704T+2T2 1 - 0.704T + 2T^{2}
3 1+3.11T+3T2 1 + 3.11T + 3T^{2}
7 1−4.06T+7T2 1 - 4.06T + 7T^{2}
11 1+0.231T+11T2 1 + 0.231T + 11T^{2}
13 1+2.35T+13T2 1 + 2.35T + 13T^{2}
19 1−7.38T+19T2 1 - 7.38T + 19T^{2}
23 1−1.87T+23T2 1 - 1.87T + 23T^{2}
29 1−4.04T+29T2 1 - 4.04T + 29T^{2}
31 1−0.130T+31T2 1 - 0.130T + 31T^{2}
37 1−5.82T+37T2 1 - 5.82T + 37T^{2}
41 1−8.10T+41T2 1 - 8.10T + 41T^{2}
43 1−3.69T+43T2 1 - 3.69T + 43T^{2}
47 1+1.30T+47T2 1 + 1.30T + 47T^{2}
53 1+8.26T+53T2 1 + 8.26T + 53T^{2}
59 1−2.29T+59T2 1 - 2.29T + 59T^{2}
61 1−8.92T+61T2 1 - 8.92T + 61T^{2}
67 1+14.5T+67T2 1 + 14.5T + 67T^{2}
71 1−11.0T+71T2 1 - 11.0T + 71T^{2}
73 1+0.471T+73T2 1 + 0.471T + 73T^{2}
79 1+4.31T+79T2 1 + 4.31T + 79T^{2}
83 1+15.4T+83T2 1 + 15.4T + 83T^{2}
89 1−3.68T+89T2 1 - 3.68T + 89T^{2}
97 1+9.00T+97T2 1 + 9.00T + 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

−7.69098731328964734914046824872, −7.21801102245718344093269707982, −6.19393456331555870520595172277, −5.57358615600817643389366537238, −5.07530673726586719572733676757, −4.65738556102620911247539313562, −4.06344893037472523954427351760, −2.78793533728328553857377158847, −1.37411412236156984834886127148, −0.68897636282364820749264082328, 0.68897636282364820749264082328, 1.37411412236156984834886127148, 2.78793533728328553857377158847, 4.06344893037472523954427351760, 4.65738556102620911247539313562, 5.07530673726586719572733676757, 5.57358615600817643389366537238, 6.19393456331555870520595172277, 7.21801102245718344093269707982, 7.69098731328964734914046824872

Graph of the ZZ-function along the critical line