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Properties

Label 2-95e2-1.1-c1-0-72
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  − 1.66·2-s − 1.15·3-s + 0.770·4-s + 1.93·6-s + 2.43·7-s + 2.04·8-s − 1.65·9-s − 5.75·11-s − 0.893·12-s − 1.59·13-s − 4.05·14-s − 4.94·16-s + 5.98·17-s + 2.75·18-s − 2.82·21-s + 9.57·22-s + 0.940·23-s − 2.37·24-s + 2.65·26-s + 5.39·27-s + 1.87·28-s − 2.61·29-s + 5.26·31-s + 4.14·32-s + 6.67·33-s − 9.96·34-s − 1.27·36-s + ⋯
L(s)  = 1  − 1.17·2-s − 0.669·3-s + 0.385·4-s + 0.788·6-s + 0.920·7-s + 0.723·8-s − 0.551·9-s − 1.73·11-s − 0.258·12-s − 0.442·13-s − 1.08·14-s − 1.23·16-s + 1.45·17-s + 0.649·18-s − 0.616·21-s + 2.04·22-s + 0.196·23-s − 0.484·24-s + 0.520·26-s + 1.03·27-s + 0.354·28-s − 0.486·29-s + 0.946·31-s + 0.732·32-s + 1.16·33-s − 1.70·34-s − 0.212·36-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 0.47682532440.4768253244
L(12)L(\frac12) ≈\approx 0.47682532440.4768253244
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+1.66T+2T2 1 + 1.66T + 2T^{2}
3 1+1.15T+3T2 1 + 1.15T + 3T^{2}
7 1−2.43T+7T2 1 - 2.43T + 7T^{2}
11 1+5.75T+11T2 1 + 5.75T + 11T^{2}
13 1+1.59T+13T2 1 + 1.59T + 13T^{2}
17 1−5.98T+17T2 1 - 5.98T + 17T^{2}
23 1−0.940T+23T2 1 - 0.940T + 23T^{2}
29 1+2.61T+29T2 1 + 2.61T + 29T^{2}
31 1−5.26T+31T2 1 - 5.26T + 31T^{2}
37 1+2.89T+37T2 1 + 2.89T + 37T^{2}
41 1−6.31T+41T2 1 - 6.31T + 41T^{2}
43 1+4.53T+43T2 1 + 4.53T + 43T^{2}
47 1+8.95T+47T2 1 + 8.95T + 47T^{2}
53 1+2.19T+53T2 1 + 2.19T + 53T^{2}
59 1−10.7T+59T2 1 - 10.7T + 59T^{2}
61 1+10.5T+61T2 1 + 10.5T + 61T^{2}
67 1−1.00T+67T2 1 - 1.00T + 67T^{2}
71 1+8.83T+71T2 1 + 8.83T + 71T^{2}
73 1−10.2T+73T2 1 - 10.2T + 73T^{2}
79 1+7.60T+79T2 1 + 7.60T + 79T^{2}
83 1+3.11T+83T2 1 + 3.11T + 83T^{2}
89 1−11.1T+89T2 1 - 11.1T + 89T^{2}
97 1−4.05T+97T2 1 - 4.05T + 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

−7.86080371179971973051156040140, −7.45328281187911444555217575427, −6.46560191328559063941042043957, −5.50609763972348593591930320635, −5.12226906379260178962265908387, −4.56226506936702915316399510193, −3.22755732048517901531965320170, −2.39842534605707709598620779330, −1.41033245981883767143394211320, −0.43993708233029008854818624015, 0.43993708233029008854818624015, 1.41033245981883767143394211320, 2.39842534605707709598620779330, 3.22755732048517901531965320170, 4.56226506936702915316399510193, 5.12226906379260178962265908387, 5.50609763972348593591930320635, 6.46560191328559063941042043957, 7.45328281187911444555217575427, 7.86080371179971973051156040140

Graph of the ZZ-function along the critical line