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Properties

Label 2-95e2-1.1-c1-0-419
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  + 2.75·2-s + 1.49·3-s + 5.60·4-s + 4.11·6-s + 2.84·7-s + 9.94·8-s − 0.774·9-s − 0.864·11-s + 8.36·12-s + 0.643·13-s + 7.85·14-s + 16.2·16-s − 3.74·17-s − 2.13·18-s + 4.24·21-s − 2.38·22-s − 0.417·23-s + 14.8·24-s + 1.77·26-s − 5.63·27-s + 15.9·28-s + 9.70·29-s − 4.93·31-s + 24.8·32-s − 1.29·33-s − 10.3·34-s − 4.34·36-s + ⋯
L(s)  = 1  + 1.95·2-s + 0.861·3-s + 2.80·4-s + 1.67·6-s + 1.07·7-s + 3.51·8-s − 0.258·9-s − 0.260·11-s + 2.41·12-s + 0.178·13-s + 2.09·14-s + 4.05·16-s − 0.907·17-s − 0.503·18-s + 0.927·21-s − 0.508·22-s − 0.0870·23-s + 3.02·24-s + 0.347·26-s − 1.08·27-s + 3.01·28-s + 1.80·29-s − 0.886·31-s + 4.39·32-s − 0.224·33-s − 1.76·34-s − 0.723·36-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 12.3138549512.31385495
L(12)L(\frac12) ≈\approx 12.3138549512.31385495
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 1−2.75T+2T2 1 - 2.75T + 2T^{2}
3 1−1.49T+3T2 1 - 1.49T + 3T^{2}
7 1−2.84T+7T2 1 - 2.84T + 7T^{2}
11 1+0.864T+11T2 1 + 0.864T + 11T^{2}
13 1−0.643T+13T2 1 - 0.643T + 13T^{2}
17 1+3.74T+17T2 1 + 3.74T + 17T^{2}
23 1+0.417T+23T2 1 + 0.417T + 23T^{2}
29 1−9.70T+29T2 1 - 9.70T + 29T^{2}
31 1+4.93T+31T2 1 + 4.93T + 31T^{2}
37 1−6.36T+37T2 1 - 6.36T + 37T^{2}
41 1−4.01T+41T2 1 - 4.01T + 41T^{2}
43 1−2.05T+43T2 1 - 2.05T + 43T^{2}
47 1−3.95T+47T2 1 - 3.95T + 47T^{2}
53 1+10.9T+53T2 1 + 10.9T + 53T^{2}
59 1+2.45T+59T2 1 + 2.45T + 59T^{2}
61 1−6.33T+61T2 1 - 6.33T + 61T^{2}
67 1+2.53T+67T2 1 + 2.53T + 67T^{2}
71 1−1.78T+71T2 1 - 1.78T + 71T^{2}
73 1−7.13T+73T2 1 - 7.13T + 73T^{2}
79 1+1.82T+79T2 1 + 1.82T + 79T^{2}
83 1−7.43T+83T2 1 - 7.43T + 83T^{2}
89 1+4.44T+89T2 1 + 4.44T + 89T^{2}
97 1+10.8T+97T2 1 + 10.8T + 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

−7.73286900395089884205998342392, −6.84976634337694986141746734805, −6.19878433548996243280833430672, −5.49619519517840036477797165948, −4.79646293763180028201366383522, −4.29229782423517606501374844596, −3.56275660414593274498909694785, −2.68683150592785329526508798278, −2.31109042727214224982455867084, −1.37072949960239552813855742296, 1.37072949960239552813855742296, 2.31109042727214224982455867084, 2.68683150592785329526508798278, 3.56275660414593274498909694785, 4.29229782423517606501374844596, 4.79646293763180028201366383522, 5.49619519517840036477797165948, 6.19878433548996243280833430672, 6.84976634337694986141746734805, 7.73286900395089884205998342392

Graph of the ZZ-function along the critical line