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Properties

Label 2-95e2-1.1-c1-0-66
Degree 22
Conductor 90259025
Sign 11
Analytic cond. 72.064972.0649
Root an. cond. 8.489108.48910
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  + 1.09·2-s − 0.379·3-s − 0.795·4-s − 0.416·6-s − 1.89·7-s − 3.06·8-s − 2.85·9-s + 0.134·11-s + 0.301·12-s + 3.51·13-s − 2.07·14-s − 1.77·16-s + 1.66·17-s − 3.13·18-s + 0.718·21-s + 0.147·22-s − 5.36·23-s + 1.16·24-s + 3.85·26-s + 2.22·27-s + 1.50·28-s − 4.97·29-s − 6.56·31-s + 4.18·32-s − 0.0509·33-s + 1.82·34-s + 2.27·36-s + ⋯
L(s)  = 1  + 0.776·2-s − 0.219·3-s − 0.397·4-s − 0.169·6-s − 0.715·7-s − 1.08·8-s − 0.952·9-s + 0.0405·11-s + 0.0871·12-s + 0.974·13-s − 0.555·14-s − 0.443·16-s + 0.402·17-s − 0.738·18-s + 0.156·21-s + 0.0314·22-s − 1.11·23-s + 0.237·24-s + 0.756·26-s + 0.427·27-s + 0.284·28-s − 0.923·29-s − 1.17·31-s + 0.740·32-s − 0.00887·33-s + 0.312·34-s + 0.378·36-s + ⋯

Functional equation

Λ(s)=(9025s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9025s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 90259025    =    52⋅1925^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 72.064972.0649
Root analytic conductor: 8.489108.48910
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9025, ( :1/2), 1)(2,\ 9025,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0441927051.044192705
L(12)L(\frac12) ≈\approx 1.0441927051.044192705
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
19 1 1
good2 1−1.09T+2T2 1 - 1.09T + 2T^{2}
3 1+0.379T+3T2 1 + 0.379T + 3T^{2}
7 1+1.89T+7T2 1 + 1.89T + 7T^{2}
11 1−0.134T+11T2 1 - 0.134T + 11T^{2}
13 1−3.51T+13T2 1 - 3.51T + 13T^{2}
17 1−1.66T+17T2 1 - 1.66T + 17T^{2}
23 1+5.36T+23T2 1 + 5.36T + 23T^{2}
29 1+4.97T+29T2 1 + 4.97T + 29T^{2}
31 1+6.56T+31T2 1 + 6.56T + 31T^{2}
37 1+1.69T+37T2 1 + 1.69T + 37T^{2}
41 1+10.6T+41T2 1 + 10.6T + 41T^{2}
43 1+8.50T+43T2 1 + 8.50T + 43T^{2}
47 1−11.1T+47T2 1 - 11.1T + 47T^{2}
53 1+0.264T+53T2 1 + 0.264T + 53T^{2}
59 1−6.89T+59T2 1 - 6.89T + 59T^{2}
61 1−9.17T+61T2 1 - 9.17T + 61T^{2}
67 1+2.95T+67T2 1 + 2.95T + 67T^{2}
71 1+1.32T+71T2 1 + 1.32T + 71T^{2}
73 1−6.34T+73T2 1 - 6.34T + 73T^{2}
79 1−1.46T+79T2 1 - 1.46T + 79T^{2}
83 1+7.44T+83T2 1 + 7.44T + 83T^{2}
89 1+9.73T+89T2 1 + 9.73T + 89T^{2}
97 1−17.4T+97T2 1 - 17.4T + 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

−7.75245173297456023729064232334, −6.73651728789592967421646594614, −6.20516162008581465675773202700, −5.52042224366305680110764744337, −5.22647917266468075162048015474, −4.01865613232690361214707742783, −3.60962161464734653995688031979, −2.98549848272529657281961996394, −1.86559124902586477304134607381, −0.42699641486266994228717594978, 0.42699641486266994228717594978, 1.86559124902586477304134607381, 2.98549848272529657281961996394, 3.60962161464734653995688031979, 4.01865613232690361214707742783, 5.22647917266468075162048015474, 5.52042224366305680110764744337, 6.20516162008581465675773202700, 6.73651728789592967421646594614, 7.75245173297456023729064232334

Graph of the ZZ-function along the critical line