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Properties

Label 2-85e2-1.1-c1-0-309
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  − 2.30·2-s + 1.87·3-s + 3.31·4-s − 4.32·6-s − 1.54·7-s − 3.03·8-s + 0.525·9-s + 4.61·11-s + 6.22·12-s + 3.51·13-s + 3.56·14-s + 0.366·16-s − 1.21·18-s − 6.62·19-s − 2.90·21-s − 10.6·22-s + 2.59·23-s − 5.70·24-s − 8.11·26-s − 4.64·27-s − 5.12·28-s − 0.979·29-s + 1.22·31-s + 5.22·32-s + 8.67·33-s + 1.74·36-s − 0.407·37-s + ⋯
L(s)  = 1  − 1.63·2-s + 1.08·3-s + 1.65·4-s − 1.76·6-s − 0.584·7-s − 1.07·8-s + 0.175·9-s + 1.39·11-s + 1.79·12-s + 0.975·13-s + 0.952·14-s + 0.0916·16-s − 0.285·18-s − 1.51·19-s − 0.633·21-s − 2.27·22-s + 0.541·23-s − 1.16·24-s − 1.59·26-s − 0.894·27-s − 0.968·28-s − 0.181·29-s + 0.219·31-s + 0.923·32-s + 1.50·33-s + 0.290·36-s − 0.0670·37-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+2.30T+2T2 1 + 2.30T + 2T^{2}
3 1−1.87T+3T2 1 - 1.87T + 3T^{2}
7 1+1.54T+7T2 1 + 1.54T + 7T^{2}
11 1−4.61T+11T2 1 - 4.61T + 11T^{2}
13 1−3.51T+13T2 1 - 3.51T + 13T^{2}
19 1+6.62T+19T2 1 + 6.62T + 19T^{2}
23 1−2.59T+23T2 1 - 2.59T + 23T^{2}
29 1+0.979T+29T2 1 + 0.979T + 29T^{2}
31 1−1.22T+31T2 1 - 1.22T + 31T^{2}
37 1+0.407T+37T2 1 + 0.407T + 37T^{2}
41 1+10.2T+41T2 1 + 10.2T + 41T^{2}
43 1+4.24T+43T2 1 + 4.24T + 43T^{2}
47 1−3.99T+47T2 1 - 3.99T + 47T^{2}
53 1+1.75T+53T2 1 + 1.75T + 53T^{2}
59 1−1.56T+59T2 1 - 1.56T + 59T^{2}
61 1+13.9T+61T2 1 + 13.9T + 61T^{2}
67 1−8.84T+67T2 1 - 8.84T + 67T^{2}
71 1+1.29T+71T2 1 + 1.29T + 71T^{2}
73 1+9.77T+73T2 1 + 9.77T + 73T^{2}
79 1−10.9T+79T2 1 - 10.9T + 79T^{2}
83 1+12.6T+83T2 1 + 12.6T + 83T^{2}
89 1+8.62T+89T2 1 + 8.62T + 89T^{2}
97 1+5.92T+97T2 1 + 5.92T + 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

−7.966067295797871977614811548414, −6.94319966583268406188638258675, −6.61546685062811788394416606953, −5.89470577278806528462047760408, −4.44788560512365127136286903666, −3.64678424917222717876681475359, −2.96200126624706109823356445082, −1.95963502762029590923170102413, −1.32778051353071546491843175399, 0, 1.32778051353071546491843175399, 1.95963502762029590923170102413, 2.96200126624706109823356445082, 3.64678424917222717876681475359, 4.44788560512365127136286903666, 5.89470577278806528462047760408, 6.61546685062811788394416606953, 6.94319966583268406188638258675, 7.966067295797871977614811548414

Graph of the ZZ-function along the critical line