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Properties

Label 2-85e2-1.1-c1-0-367
Degree 22
Conductor 72257225
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.15·2-s + 2.05·3-s − 0.677·4-s + 2.36·6-s − 0.375·7-s − 3.07·8-s + 1.22·9-s − 0.0553·11-s − 1.39·12-s − 0.388·13-s − 0.431·14-s − 2.18·16-s + 1.40·18-s + 1.76·19-s − 0.771·21-s − 0.0636·22-s + 1.02·23-s − 6.32·24-s − 0.446·26-s − 3.64·27-s + 0.254·28-s + 8.45·29-s − 6.05·31-s + 3.64·32-s − 0.113·33-s − 0.829·36-s − 9.49·37-s + ⋯
L(s)  = 1  + 0.813·2-s + 1.18·3-s − 0.338·4-s + 0.965·6-s − 0.141·7-s − 1.08·8-s + 0.408·9-s − 0.0166·11-s − 0.401·12-s − 0.107·13-s − 0.115·14-s − 0.546·16-s + 0.332·18-s + 0.405·19-s − 0.168·21-s − 0.0135·22-s + 0.214·23-s − 1.29·24-s − 0.0876·26-s − 0.702·27-s + 0.0480·28-s + 1.56·29-s − 1.08·31-s + 0.643·32-s − 0.0198·33-s − 0.138·36-s − 1.56·37-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: −1-1
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: 11
Selberg data: (2, 7225, ( :1/2), −1)(2,\ 7225,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−1.15T+2T2 1 - 1.15T + 2T^{2}
3 1−2.05T+3T2 1 - 2.05T + 3T^{2}
7 1+0.375T+7T2 1 + 0.375T + 7T^{2}
11 1+0.0553T+11T2 1 + 0.0553T + 11T^{2}
13 1+0.388T+13T2 1 + 0.388T + 13T^{2}
19 1−1.76T+19T2 1 - 1.76T + 19T^{2}
23 1−1.02T+23T2 1 - 1.02T + 23T^{2}
29 1−8.45T+29T2 1 - 8.45T + 29T^{2}
31 1+6.05T+31T2 1 + 6.05T + 31T^{2}
37 1+9.49T+37T2 1 + 9.49T + 37T^{2}
41 1+7.67T+41T2 1 + 7.67T + 41T^{2}
43 1+6.75T+43T2 1 + 6.75T + 43T^{2}
47 1−1.84T+47T2 1 - 1.84T + 47T^{2}
53 1+9.09T+53T2 1 + 9.09T + 53T^{2}
59 1−13.5T+59T2 1 - 13.5T + 59T^{2}
61 1+5.81T+61T2 1 + 5.81T + 61T^{2}
67 1+14.4T+67T2 1 + 14.4T + 67T^{2}
71 1+1.12T+71T2 1 + 1.12T + 71T^{2}
73 1+2.19T+73T2 1 + 2.19T + 73T^{2}
79 1−3.94T+79T2 1 - 3.94T + 79T^{2}
83 1−9.44T+83T2 1 - 9.44T + 83T^{2}
89 1−2.61T+89T2 1 - 2.61T + 89T^{2}
97 1+5.25T+97T2 1 + 5.25T + 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

−7.63884075259623633446756059232, −6.84229509382842268278412562757, −6.12160096022818653281178002360, −5.19225434719845765737270667836, −4.75390344054882532421949774437, −3.69319483653592712303226492912, −3.32716756996508499802703298758, −2.63074677926050812618688573133, −1.58470737411709746085684425796, 0, 1.58470737411709746085684425796, 2.63074677926050812618688573133, 3.32716756996508499802703298758, 3.69319483653592712303226492912, 4.75390344054882532421949774437, 5.19225434719845765737270667836, 6.12160096022818653281178002360, 6.84229509382842268278412562757, 7.63884075259623633446756059232

Graph of the ZZ-function along the critical line