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Properties

Label 2-85e2-1.1-c1-0-142
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  − 1.99·2-s − 2.94·3-s + 1.96·4-s + 5.85·6-s − 3.47·7-s + 0.0709·8-s + 5.65·9-s − 3.24·11-s − 5.77·12-s + 3.53·13-s + 6.92·14-s − 4.07·16-s − 11.2·18-s + 1.42·19-s + 10.2·21-s + 6.46·22-s + 7.40·23-s − 0.208·24-s − 7.03·26-s − 7.79·27-s − 6.83·28-s − 10.4·29-s − 1.74·31-s + 7.96·32-s + 9.54·33-s + 11.0·36-s − 3.36·37-s + ⋯
L(s)  = 1  − 1.40·2-s − 1.69·3-s + 0.982·4-s + 2.39·6-s − 1.31·7-s + 0.0251·8-s + 1.88·9-s − 0.978·11-s − 1.66·12-s + 0.979·13-s + 1.85·14-s − 1.01·16-s − 2.65·18-s + 0.326·19-s + 2.23·21-s + 1.37·22-s + 1.54·23-s − 0.0426·24-s − 1.37·26-s − 1.50·27-s − 1.29·28-s − 1.93·29-s − 0.313·31-s + 1.40·32-s + 1.66·33-s + 1.84·36-s − 0.553·37-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+1.99T+2T2 1 + 1.99T + 2T^{2}
3 1+2.94T+3T2 1 + 2.94T + 3T^{2}
7 1+3.47T+7T2 1 + 3.47T + 7T^{2}
11 1+3.24T+11T2 1 + 3.24T + 11T^{2}
13 1−3.53T+13T2 1 - 3.53T + 13T^{2}
19 1−1.42T+19T2 1 - 1.42T + 19T^{2}
23 1−7.40T+23T2 1 - 7.40T + 23T^{2}
29 1+10.4T+29T2 1 + 10.4T + 29T^{2}
31 1+1.74T+31T2 1 + 1.74T + 31T^{2}
37 1+3.36T+37T2 1 + 3.36T + 37T^{2}
41 1−3.91T+41T2 1 - 3.91T + 41T^{2}
43 1+6.68T+43T2 1 + 6.68T + 43T^{2}
47 1+1.75T+47T2 1 + 1.75T + 47T^{2}
53 1+2.10T+53T2 1 + 2.10T + 53T^{2}
59 1−6.95T+59T2 1 - 6.95T + 59T^{2}
61 1−7.31T+61T2 1 - 7.31T + 61T^{2}
67 1+0.934T+67T2 1 + 0.934T + 67T^{2}
71 1+16.5T+71T2 1 + 16.5T + 71T^{2}
73 1−2.58T+73T2 1 - 2.58T + 73T^{2}
79 1−11.6T+79T2 1 - 11.6T + 79T^{2}
83 1+6.23T+83T2 1 + 6.23T + 83T^{2}
89 1−10.6T+89T2 1 - 10.6T + 89T^{2}
97 1+4.47T+97T2 1 + 4.47T + 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

−7.40836021832427240498702933003, −6.94563379591907070253413930420, −6.36022327203892939932869475562, −5.57844861588883252675943559444, −5.08369525041850112098523278722, −3.97700771719244411000965456597, −3.03735315044903991897585737769, −1.72574691163623699570083992809, −0.74643705496915680529712658322, 0, 0.74643705496915680529712658322, 1.72574691163623699570083992809, 3.03735315044903991897585737769, 3.97700771719244411000965456597, 5.08369525041850112098523278722, 5.57844861588883252675943559444, 6.36022327203892939932869475562, 6.94563379591907070253413930420, 7.40836021832427240498702933003

Graph of the ZZ-function along the critical line