↑

Properties

Label 2-88e2-1.1-c1-0-96
Degree 22
Conductor 77447744
Sign −1-1
Analytic cond. 61.836161.8361
Root an. cond. 7.863597.86359
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  − 2.23·3-s − 5-s − 2·7-s + 2.00·9-s + 4.47·13-s + 2.23·15-s − 4.47·17-s + 6·19-s + 4.47·21-s + 2.23·23-s − 4·25-s + 2.23·27-s − 8.94·29-s + 2.23·31-s + 2·35-s − 7·37-s − 10.0·39-s − 4.47·41-s − 4·43-s − 2.00·45-s − 8.94·47-s − 3·49-s + 10.0·51-s + 6·53-s − 13.4·57-s + 11.1·59-s + 8.94·61-s + ⋯
L(s)  = 1  − 1.29·3-s − 0.447·5-s − 0.755·7-s + 0.666·9-s + 1.24·13-s + 0.577·15-s − 1.08·17-s + 1.37·19-s + 0.975·21-s + 0.466·23-s − 0.800·25-s + 0.430·27-s − 1.66·29-s + 0.401·31-s + 0.338·35-s − 1.15·37-s − 1.60·39-s − 0.698·41-s − 0.609·43-s − 0.298·45-s − 1.30·47-s − 0.428·49-s + 1.40·51-s + 0.824·53-s − 1.77·57-s + 1.45·59-s + 1.14·61-s + ⋯

Functional equation

Λ(s)=(7744s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7744s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 77447744    =    26⋅1122^{6} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 61.836161.8361
Root analytic conductor: 7.863597.86359
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7744, ( :1/2), −1)(2,\ 7744,\ (\ :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
11 1 1
good3 1+2.23T+3T2 1 + 2.23T + 3T^{2}
5 1+T+5T2 1 + T + 5T^{2}
7 1+2T+7T2 1 + 2T + 7T^{2}
13 1−4.47T+13T2 1 - 4.47T + 13T^{2}
17 1+4.47T+17T2 1 + 4.47T + 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1−2.23T+23T2 1 - 2.23T + 23T^{2}
29 1+8.94T+29T2 1 + 8.94T + 29T^{2}
31 1−2.23T+31T2 1 - 2.23T + 31T^{2}
37 1+7T+37T2 1 + 7T + 37T^{2}
41 1+4.47T+41T2 1 + 4.47T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1+8.94T+47T2 1 + 8.94T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1−11.1T+59T2 1 - 11.1T + 59T^{2}
61 1−8.94T+61T2 1 - 8.94T + 61T^{2}
67 1−11.1T+67T2 1 - 11.1T + 67T^{2}
71 1−6.70T+71T2 1 - 6.70T + 71T^{2}
73 1−13.4T+73T2 1 - 13.4T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−16T+83T2 1 - 16T + 83T^{2}
89 1+9T+89T2 1 + 9T + 89T^{2}
97 1−13T+97T2 1 - 13T + 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

−7.28611596361409588870020032068, −6.62959227496604694603637158077, −6.25118139849277202438466544731, −5.33066739748235138209750570993, −5.01310558201113139047297274281, −3.64684345307078130715520655152, −3.58241100535013479123316662261, −2.11038101479190193356541012198, −0.933007966542278322497195998809, 0, 0.933007966542278322497195998809, 2.11038101479190193356541012198, 3.58241100535013479123316662261, 3.64684345307078130715520655152, 5.01310558201113139047297274281, 5.33066739748235138209750570993, 6.25118139849277202438466544731, 6.62959227496604694603637158077, 7.28611596361409588870020032068

Graph of the ZZ-function along the critical line