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Properties

Label 2-85e2-1.1-c1-0-101
Degree 22
Conductor 72257225
Sign 11
Analytic cond. 57.691957.6919
Root an. cond. 7.595517.59551
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s + 6-s − 7-s − 3·8-s − 2·9-s + 4·11-s − 12-s − 13-s − 14-s − 16-s − 2·18-s − 6·19-s − 21-s + 4·22-s − 3·24-s − 26-s − 5·27-s + 28-s + 7·31-s + 5·32-s + 4·33-s + 2·36-s + 4·37-s − 6·38-s − 39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s − 1.06·8-s − 2/3·9-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 1/4·16-s − 0.471·18-s − 1.37·19-s − 0.218·21-s + 0.852·22-s − 0.612·24-s − 0.196·26-s − 0.962·27-s + 0.188·28-s + 1.25·31-s + 0.883·32-s + 0.696·33-s + 1/3·36-s + 0.657·37-s − 0.973·38-s − 0.160·39-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: yes
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 2.2266936722.226693672
L(12)L(\frac12) ≈\approx 2.2266936722.226693672
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−T+pT2 1 - T + p T^{2}
3 1−T+pT2 1 - T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+7T+pT2 1 + 7 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1−11T+pT2 1 - 11 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−16T+pT2 1 - 16 T + p T^{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

−8.137336175370570455158370286211, −7.11506758484051445834837725606, −6.18382323461402075157966073920, −6.03879953647401353841158787955, −4.88611047485524815987471444871, −4.28031132843975574213598865876, −3.62547166449923999392099280690, −2.92386518865550189156320772474, −2.10237647492739208778648010240, −0.64079892155178707280465812390, 0.64079892155178707280465812390, 2.10237647492739208778648010240, 2.92386518865550189156320772474, 3.62547166449923999392099280690, 4.28031132843975574213598865876, 4.88611047485524815987471444871, 6.03879953647401353841158787955, 6.18382323461402075157966073920, 7.11506758484051445834837725606, 8.137336175370570455158370286211

Graph of the ZZ-function along the critical line