↑

Properties

Label 2-85e2-1.1-c1-0-223
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.74·2-s + 0.598·3-s + 1.03·4-s − 1.04·6-s − 1.03·7-s + 1.68·8-s − 2.64·9-s + 0.0764·11-s + 0.620·12-s − 2.86·13-s + 1.80·14-s − 4.99·16-s + 4.60·18-s + 4.19·19-s − 0.621·21-s − 0.133·22-s + 5.95·23-s + 1.00·24-s + 4.98·26-s − 3.37·27-s − 1.07·28-s − 0.0884·29-s − 2.42·31-s + 5.34·32-s + 0.0457·33-s − 2.73·36-s − 9.51·37-s + ⋯
L(s)  = 1  − 1.23·2-s + 0.345·3-s + 0.517·4-s − 0.425·6-s − 0.392·7-s + 0.594·8-s − 0.880·9-s + 0.0230·11-s + 0.178·12-s − 0.793·13-s + 0.483·14-s − 1.24·16-s + 1.08·18-s + 0.961·19-s − 0.135·21-s − 0.0284·22-s + 1.24·23-s + 0.205·24-s + 0.977·26-s − 0.650·27-s − 0.203·28-s − 0.0164·29-s − 0.435·31-s + 0.945·32-s + 0.00796·33-s − 0.455·36-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+1.74T+2T2 1 + 1.74T + 2T^{2}
3 1−0.598T+3T2 1 - 0.598T + 3T^{2}
7 1+1.03T+7T2 1 + 1.03T + 7T^{2}
11 1−0.0764T+11T2 1 - 0.0764T + 11T^{2}
13 1+2.86T+13T2 1 + 2.86T + 13T^{2}
19 1−4.19T+19T2 1 - 4.19T + 19T^{2}
23 1−5.95T+23T2 1 - 5.95T + 23T^{2}
29 1+0.0884T+29T2 1 + 0.0884T + 29T^{2}
31 1+2.42T+31T2 1 + 2.42T + 31T^{2}
37 1+9.51T+37T2 1 + 9.51T + 37T^{2}
41 1−8.37T+41T2 1 - 8.37T + 41T^{2}
43 1+0.866T+43T2 1 + 0.866T + 43T^{2}
47 1−8.33T+47T2 1 - 8.33T + 47T^{2}
53 1−4.09T+53T2 1 - 4.09T + 53T^{2}
59 1+9.16T+59T2 1 + 9.16T + 59T^{2}
61 1+0.288T+61T2 1 + 0.288T + 61T^{2}
67 1−12.8T+67T2 1 - 12.8T + 67T^{2}
71 1+13.2T+71T2 1 + 13.2T + 71T^{2}
73 1−6.54T+73T2 1 - 6.54T + 73T^{2}
79 1−10.4T+79T2 1 - 10.4T + 79T^{2}
83 1−9.70T+83T2 1 - 9.70T + 83T^{2}
89 1−0.106T+89T2 1 - 0.106T + 89T^{2}
97 1+8.56T+97T2 1 + 8.56T + 97T^{2}
show more
show less
   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

−7.57509174528457394984270785460, −7.27754977926065818306541520581, −6.42285708455959258008478125668, −5.42408332677987315037096629999, −4.87515460727177752847276139194, −3.74729457157198604399940156233, −2.93528865007478104664663021416, −2.16695100387509996963540852323, −1.03099152099989864544856515130, 0, 1.03099152099989864544856515130, 2.16695100387509996963540852323, 2.93528865007478104664663021416, 3.74729457157198604399940156233, 4.87515460727177752847276139194, 5.42408332677987315037096629999, 6.42285708455959258008478125668, 7.27754977926065818306541520581, 7.57509174528457394984270785460

Graph of the ZZ-function along the critical line