↑

Properties

Label 2-69e2-1.1-c1-0-187
Degree 22
Conductor 47614761
Sign −1-1
Analytic cond. 38.016738.0167
Root an. cond. 6.165776.16577
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.51·2-s + 4.31·4-s − 4.05·5-s + 1.37·7-s + 5.82·8-s − 10.1·10-s + 0.0273·11-s − 0.627·13-s + 3.46·14-s + 6.00·16-s − 4.30·17-s − 5.00·19-s − 17.5·20-s + 0.0686·22-s + 11.4·25-s − 1.57·26-s + 5.94·28-s − 3.69·29-s − 0.482·31-s + 3.43·32-s − 10.8·34-s − 5.58·35-s + 2.20·37-s − 12.5·38-s − 23.6·40-s − 10.0·41-s + 3.28·43-s + ⋯
L(s)  = 1  + 1.77·2-s + 2.15·4-s − 1.81·5-s + 0.520·7-s + 2.05·8-s − 3.22·10-s + 0.00823·11-s − 0.173·13-s + 0.924·14-s + 1.50·16-s − 1.04·17-s − 1.14·19-s − 3.91·20-s + 0.0146·22-s + 2.28·25-s − 0.309·26-s + 1.12·28-s − 0.686·29-s − 0.0866·31-s + 0.607·32-s − 1.85·34-s − 0.943·35-s + 0.362·37-s − 2.03·38-s − 3.73·40-s − 1.56·41-s + 0.500·43-s + ⋯

Functional equation

Λ(s)=(4761s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4761s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47614761    =    32⋅2323^{2} \cdot 23^{2}
Sign: −1-1
Analytic conductor: 38.016738.0167
Root analytic conductor: 6.165776.16577
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4761, ( :1/2), −1)(2,\ 4761,\ (\ :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)
bad3 1 1
23 1 1
good2 1−2.51T+2T2 1 - 2.51T + 2T^{2}
5 1+4.05T+5T2 1 + 4.05T + 5T^{2}
7 1−1.37T+7T2 1 - 1.37T + 7T^{2}
11 1−0.0273T+11T2 1 - 0.0273T + 11T^{2}
13 1+0.627T+13T2 1 + 0.627T + 13T^{2}
17 1+4.30T+17T2 1 + 4.30T + 17T^{2}
19 1+5.00T+19T2 1 + 5.00T + 19T^{2}
29 1+3.69T+29T2 1 + 3.69T + 29T^{2}
31 1+0.482T+31T2 1 + 0.482T + 31T^{2}
37 1−2.20T+37T2 1 - 2.20T + 37T^{2}
41 1+10.0T+41T2 1 + 10.0T + 41T^{2}
43 1−3.28T+43T2 1 - 3.28T + 43T^{2}
47 1+8.98T+47T2 1 + 8.98T + 47T^{2}
53 1−1.83T+53T2 1 - 1.83T + 53T^{2}
59 1−4.17T+59T2 1 - 4.17T + 59T^{2}
61 1+4.62T+61T2 1 + 4.62T + 61T^{2}
67 1+4.42T+67T2 1 + 4.42T + 67T^{2}
71 1−7.22T+71T2 1 - 7.22T + 71T^{2}
73 1−5.97T+73T2 1 - 5.97T + 73T^{2}
79 1+1.50T+79T2 1 + 1.50T + 79T^{2}
83 1+8.56T+83T2 1 + 8.56T + 83T^{2}
89 1−13.3T+89T2 1 - 13.3T + 89T^{2}
97 1+2.16T+97T2 1 + 2.16T + 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

−7.83780726767064636703930181195, −6.84564943766112887328952193217, −6.61793576134925831295634803091, −5.42119517708182603264191189878, −4.70370491055110911500031204266, −4.23213875037214924324199902460, −3.65505455887020547546744618057, −2.82109756727689480773476315781, −1.81009402721287314188056961604, 0, 1.81009402721287314188056961604, 2.82109756727689480773476315781, 3.65505455887020547546744618057, 4.23213875037214924324199902460, 4.70370491055110911500031204266, 5.42119517708182603264191189878, 6.61793576134925831295634803091, 6.84564943766112887328952193217, 7.83780726767064636703930181195

Graph of the ZZ-function along the critical line