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Properties

Label 2-85e2-1.1-c1-0-141
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  + 1.22·2-s − 2.38·3-s − 0.511·4-s − 2.91·6-s + 1.22·7-s − 3.06·8-s + 2.69·9-s + 5.09·11-s + 1.22·12-s + 2.60·13-s + 1.48·14-s − 2.71·16-s + 3.28·18-s + 8.51·19-s − 2.91·21-s + 6.21·22-s − 2.89·23-s + 7.31·24-s + 3.18·26-s + 0.732·27-s − 0.623·28-s + 5.49·29-s + 3.18·31-s + 2.81·32-s − 12.1·33-s − 1.37·36-s + 3.09·37-s + ⋯
L(s)  = 1  + 0.862·2-s − 1.37·3-s − 0.255·4-s − 1.18·6-s + 0.461·7-s − 1.08·8-s + 0.897·9-s + 1.53·11-s + 0.352·12-s + 0.723·13-s + 0.397·14-s − 0.678·16-s + 0.774·18-s + 1.95·19-s − 0.635·21-s + 1.32·22-s − 0.603·23-s + 1.49·24-s + 0.623·26-s + 0.140·27-s − 0.117·28-s + 1.01·29-s + 0.571·31-s + 0.497·32-s − 2.11·33-s − 0.229·36-s + 0.508·37-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 1.9807038701.980703870
L(12)L(\frac12) ≈\approx 1.9807038701.980703870
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 1−1.22T+2T2 1 - 1.22T + 2T^{2}
3 1+2.38T+3T2 1 + 2.38T + 3T^{2}
7 1−1.22T+7T2 1 - 1.22T + 7T^{2}
11 1−5.09T+11T2 1 - 5.09T + 11T^{2}
13 1−2.60T+13T2 1 - 2.60T + 13T^{2}
19 1−8.51T+19T2 1 - 8.51T + 19T^{2}
23 1+2.89T+23T2 1 + 2.89T + 23T^{2}
29 1−5.49T+29T2 1 - 5.49T + 29T^{2}
31 1−3.18T+31T2 1 - 3.18T + 31T^{2}
37 1−3.09T+37T2 1 - 3.09T + 37T^{2}
41 1+0.181T+41T2 1 + 0.181T + 41T^{2}
43 1+6.70T+43T2 1 + 6.70T + 43T^{2}
47 1+8.71T+47T2 1 + 8.71T + 47T^{2}
53 1−0.167T+53T2 1 - 0.167T + 53T^{2}
59 1+13.7T+59T2 1 + 13.7T + 59T^{2}
61 1−0.293T+61T2 1 - 0.293T + 61T^{2}
67 1+1.56T+67T2 1 + 1.56T + 67T^{2}
71 1−6.93T+71T2 1 - 6.93T + 71T^{2}
73 1−8.04T+73T2 1 - 8.04T + 73T^{2}
79 1+0.0888T+79T2 1 + 0.0888T + 79T^{2}
83 1−14.8T+83T2 1 - 14.8T + 83T^{2}
89 1+2.79T+89T2 1 + 2.79T + 89T^{2}
97 1−3.54T+97T2 1 - 3.54T + 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

−7.87426307546264987740899511404, −6.73148502149467945532010113634, −6.33969525647206687883409990802, −5.79467782302637750941248046226, −4.95793909633553683690286215767, −4.66419318917752446092193352069, −3.72767532685911194390368713903, −3.11705478731126791805328075115, −1.49949089497666042379444493184, −0.74497121649527636883217597998, 0.74497121649527636883217597998, 1.49949089497666042379444493184, 3.11705478731126791805328075115, 3.72767532685911194390368713903, 4.66419318917752446092193352069, 4.95793909633553683690286215767, 5.79467782302637750941248046226, 6.33969525647206687883409990802, 6.73148502149467945532010113634, 7.87426307546264987740899511404

Graph of the ZZ-function along the critical line