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Properties

Label 2-85e2-1.1-c1-0-130
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  − 0.567·2-s − 2.88·3-s − 1.67·4-s + 1.63·6-s − 5.00·7-s + 2.08·8-s + 5.32·9-s + 4.17·11-s + 4.84·12-s + 0.657·13-s + 2.84·14-s + 2.16·16-s − 3.02·18-s − 5.62·19-s + 14.4·21-s − 2.37·22-s − 4.27·23-s − 6.02·24-s − 0.373·26-s − 6.71·27-s + 8.40·28-s − 5.59·29-s + 0.143·31-s − 5.40·32-s − 12.0·33-s − 8.93·36-s − 4.88·37-s + ⋯
L(s)  = 1  − 0.401·2-s − 1.66·3-s − 0.838·4-s + 0.669·6-s − 1.89·7-s + 0.738·8-s + 1.77·9-s + 1.25·11-s + 1.39·12-s + 0.182·13-s + 0.760·14-s + 0.542·16-s − 0.713·18-s − 1.28·19-s + 3.15·21-s − 0.505·22-s − 0.892·23-s − 1.23·24-s − 0.0732·26-s − 1.29·27-s + 1.58·28-s − 1.03·29-s + 0.0258·31-s − 0.956·32-s − 2.09·33-s − 1.48·36-s − 0.803·37-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+0.567T+2T2 1 + 0.567T + 2T^{2}
3 1+2.88T+3T2 1 + 2.88T + 3T^{2}
7 1+5.00T+7T2 1 + 5.00T + 7T^{2}
11 1−4.17T+11T2 1 - 4.17T + 11T^{2}
13 1−0.657T+13T2 1 - 0.657T + 13T^{2}
19 1+5.62T+19T2 1 + 5.62T + 19T^{2}
23 1+4.27T+23T2 1 + 4.27T + 23T^{2}
29 1+5.59T+29T2 1 + 5.59T + 29T^{2}
31 1−0.143T+31T2 1 - 0.143T + 31T^{2}
37 1+4.88T+37T2 1 + 4.88T + 37T^{2}
41 1+1.55T+41T2 1 + 1.55T + 41T^{2}
43 1+2.97T+43T2 1 + 2.97T + 43T^{2}
47 1+11.4T+47T2 1 + 11.4T + 47T^{2}
53 1−13.3T+53T2 1 - 13.3T + 53T^{2}
59 1−10.1T+59T2 1 - 10.1T + 59T^{2}
61 1+0.961T+61T2 1 + 0.961T + 61T^{2}
67 1−0.747T+67T2 1 - 0.747T + 67T^{2}
71 1−0.0687T+71T2 1 - 0.0687T + 71T^{2}
73 1+1.46T+73T2 1 + 1.46T + 73T^{2}
79 1−6.72T+79T2 1 - 6.72T + 79T^{2}
83 1−6.20T+83T2 1 - 6.20T + 83T^{2}
89 1−10.9T+89T2 1 - 10.9T + 89T^{2}
97 1−7.76T+97T2 1 - 7.76T + 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

−7.28720515462880064215200515436, −6.64983015903704041987460493020, −6.24507448900286901777486608976, −5.64962478991780222396187505184, −4.80025967630087028412037389536, −3.88936611252361027400858986371, −3.62296912796068090782005192178, −1.90584200732477861356459808590, −0.71896903901915469068374365289, 0, 0.71896903901915469068374365289, 1.90584200732477861356459808590, 3.62296912796068090782005192178, 3.88936611252361027400858986371, 4.80025967630087028412037389536, 5.64962478991780222396187505184, 6.24507448900286901777486608976, 6.64983015903704041987460493020, 7.28720515462880064215200515436

Graph of the ZZ-function along the critical line