↑

Properties

Label 2-5408-1.1-c1-0-132
Degree 22
Conductor 54085408
Sign −1-1
Analytic cond. 43.183043.1830
Root an. cond. 6.571386.57138
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  + 2.60·7-s − 3·9-s + 6.60·11-s − 7.21·17-s + 1.39·19-s − 5·25-s − 7.21·29-s − 10.6·31-s − 5.39·47-s − 0.211·49-s − 2·53-s − 11.8·59-s + 6·61-s − 7.81·63-s − 14.6·67-s + 15.8·71-s + 17.2·77-s + 9·81-s + 3.81·83-s − 19.8·99-s − 14·101-s + 7.21·113-s − 18.7·119-s + ⋯
L(s)  = 1  + 0.984·7-s − 9-s + 1.99·11-s − 1.74·17-s + 0.319·19-s − 25-s − 1.33·29-s − 1.90·31-s − 0.786·47-s − 0.0301·49-s − 0.274·53-s − 1.53·59-s + 0.768·61-s − 0.984·63-s − 1.78·67-s + 1.87·71-s + 1.96·77-s + 81-s + 0.418·83-s − 1.99·99-s − 1.39·101-s + 0.678·113-s − 1.72·119-s + ⋯

Functional equation

Λ(s)=(5408s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5408s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 54085408    =    25⋅1322^{5} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 43.183043.1830
Root analytic conductor: 6.571386.57138
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5408, ( :1/2), −1)(2,\ 5408,\ (\ :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)
bad2 1 1
13 1 1
good3 1+3T2 1 + 3T^{2}
5 1+5T2 1 + 5T^{2}
7 1−2.60T+7T2 1 - 2.60T + 7T^{2}
11 1−6.60T+11T2 1 - 6.60T + 11T^{2}
17 1+7.21T+17T2 1 + 7.21T + 17T^{2}
19 1−1.39T+19T2 1 - 1.39T + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+7.21T+29T2 1 + 7.21T + 29T^{2}
31 1+10.6T+31T2 1 + 10.6T + 31T^{2}
37 1+37T2 1 + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+5.39T+47T2 1 + 5.39T + 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1+11.8T+59T2 1 + 11.8T + 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1+14.6T+67T2 1 + 14.6T + 67T^{2}
71 1−15.8T+71T2 1 - 15.8T + 71T^{2}
73 1+73T2 1 + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−3.81T+83T2 1 - 3.81T + 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+97T2 1 + 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

−7.85657080605858470662776653455, −7.08839238731151107299297754803, −6.35464806187057239310107954085, −5.71524195192442254381361816615, −4.85840413614407504474505684834, −4.05946404167772737791609805592, −3.45343090505247130253682627467, −2.10866697353315047337281727212, −1.55926025733303843746617125086, 0, 1.55926025733303843746617125086, 2.10866697353315047337281727212, 3.45343090505247130253682627467, 4.05946404167772737791609805592, 4.85840413614407504474505684834, 5.71524195192442254381361816615, 6.35464806187057239310107954085, 7.08839238731151107299297754803, 7.85657080605858470662776653455

Graph of the ZZ-function along the critical line