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Properties

Label 2-85e2-1.1-c1-0-12
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  + 1.57·2-s − 2.87·3-s + 0.494·4-s − 4.53·6-s − 1.42·7-s − 2.37·8-s + 5.24·9-s − 0.0740·11-s − 1.42·12-s − 5.70·13-s − 2.24·14-s − 4.74·16-s + 8.29·18-s − 4.90·19-s + 4.08·21-s − 0.116·22-s − 3.88·23-s + 6.82·24-s − 9.00·26-s − 6.46·27-s − 0.702·28-s − 5.91·29-s − 0.388·31-s − 2.73·32-s + 0.212·33-s + 2.59·36-s − 9.91·37-s + ⋯
L(s)  = 1  + 1.11·2-s − 1.65·3-s + 0.247·4-s − 1.85·6-s − 0.536·7-s − 0.840·8-s + 1.74·9-s − 0.0223·11-s − 0.410·12-s − 1.58·13-s − 0.599·14-s − 1.18·16-s + 1.95·18-s − 1.12·19-s + 0.890·21-s − 0.0249·22-s − 0.809·23-s + 1.39·24-s − 1.76·26-s − 1.24·27-s − 0.132·28-s − 1.09·29-s − 0.0697·31-s − 0.484·32-s + 0.0370·33-s + 0.432·36-s − 1.62·37-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 0.21558998130.2155899813
L(12)L(\frac12) ≈\approx 0.21558998130.2155899813
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 1−1.57T+2T2 1 - 1.57T + 2T^{2}
3 1+2.87T+3T2 1 + 2.87T + 3T^{2}
7 1+1.42T+7T2 1 + 1.42T + 7T^{2}
11 1+0.0740T+11T2 1 + 0.0740T + 11T^{2}
13 1+5.70T+13T2 1 + 5.70T + 13T^{2}
19 1+4.90T+19T2 1 + 4.90T + 19T^{2}
23 1+3.88T+23T2 1 + 3.88T + 23T^{2}
29 1+5.91T+29T2 1 + 5.91T + 29T^{2}
31 1+0.388T+31T2 1 + 0.388T + 31T^{2}
37 1+9.91T+37T2 1 + 9.91T + 37T^{2}
41 1−6.61T+41T2 1 - 6.61T + 41T^{2}
43 1−6.94T+43T2 1 - 6.94T + 43T^{2}
47 1+5.70T+47T2 1 + 5.70T + 47T^{2}
53 1+0.0216T+53T2 1 + 0.0216T + 53T^{2}
59 1+2T+59T2 1 + 2T + 59T^{2}
61 1−3.47T+61T2 1 - 3.47T + 61T^{2}
67 1+6.71T+67T2 1 + 6.71T + 67T^{2}
71 1+3.84T+71T2 1 + 3.84T + 71T^{2}
73 1+13.5T+73T2 1 + 13.5T + 73T^{2}
79 1−1.05T+79T2 1 - 1.05T + 79T^{2}
83 1+11.8T+83T2 1 + 11.8T + 83T^{2}
89 1+1.99T+89T2 1 + 1.99T + 89T^{2}
97 1+5.34T+97T2 1 + 5.34T + 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

−7.47502191902226990355002726052, −6.93565405947524067888264592973, −6.18329957080066967936330491918, −5.77471836637819899294204594838, −5.11302811207022178907863060787, −4.50866565558689902509461754978, −3.95701940656231920615422188950, −2.89678617273194153296097916513, −1.88918383316780615893028055169, −0.20311218518310464062935841382, 0.20311218518310464062935841382, 1.88918383316780615893028055169, 2.89678617273194153296097916513, 3.95701940656231920615422188950, 4.50866565558689902509461754978, 5.11302811207022178907863060787, 5.77471836637819899294204594838, 6.18329957080066967936330491918, 6.93565405947524067888264592973, 7.47502191902226990355002726052

Graph of the ZZ-function along the critical line