↑

Properties

Label 4-588e2-1.1-c1e2-0-30
Degree 44
Conductor 345744345744
Sign 11
Analytic cond. 22.044922.0449
Root an. cond. 2.166842.16684
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s − 2·11-s + 8·13-s + 2·15-s + 6·17-s + 8·19-s + 6·23-s + 5·25-s − 27-s − 20·29-s + 4·31-s − 2·33-s − 6·37-s + 8·39-s + 12·41-s + 8·43-s + 8·47-s + 6·51-s − 2·53-s − 4·55-s + 8·57-s − 4·59-s − 8·61-s + 16·65-s + 8·67-s + 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.603·11-s + 2.21·13-s + 0.516·15-s + 1.45·17-s + 1.83·19-s + 1.25·23-s + 25-s − 0.192·27-s − 3.71·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s + 1.28·39-s + 1.87·41-s + 1.21·43-s + 1.16·47-s + 0.840·51-s − 0.274·53-s − 0.539·55-s + 1.05·57-s − 0.520·59-s − 1.02·61-s + 1.98·65-s + 0.977·67-s + 0.722·69-s + ⋯

Functional equation

Λ(s)=(345744s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(345744s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 345744345744    =    24⋅32⋅742^{4} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 22.044922.0449
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 345744, ( :1/2,1/2), 1)(4,\ 345744,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2786653513.278665351
L(12)L(\frac12) ≈\approx 3.2786653513.278665351
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C2C_2 1−T+T2 1 - T + T^{2}
7 1 1
good5C22C_2^2 1−2T−T2−2pT3+p2T4 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+2T−7T2+2pT3+p2T4 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
17C22C_2^2 1−6T+19T2−6pT3+p2T4 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4}
19C2C_2 (1−7T+pT2)(1−T+pT2) ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} )
23C22C_2^2 1−6T+13T2−6pT3+p2T4 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4}
29C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
31C2C_2 (1−11T+pT2)(1+7T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C22C_2^2 1+6T−T2+6pT3+p2T4 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1−8T+17T2−8pT3+p2T4 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+2T−49T2+2pT3+p2T4 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+4T−43T2+4pT3+p2T4 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+8T+3T2+8pT3+p2T4 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−8T−3T2−8pT3+p2T4 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4}
71C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
73C22C_2^2 1−4T−57T2−4pT3+p2T4 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4}
79C2C_2 (1−13T+pT2)(1+17T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} )
83C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
89C22C_2^2 1+14T+107T2+14pT3+p2T4 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4}
97C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
show more
show less
   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.73965459330473160920261018699, −10.72778583873254232817321505124, −9.731319955720249993534157848566, −9.688946276586426160804765139668, −9.142962783455452026301481432854, −8.835877260878964498105470781262, −8.412426757454182326040029176675, −7.63984826694279960957230548952, −7.45629129092727535273265018336, −7.14265581430412680312797508121, −6.11892232742314159672942746422, −5.85860392646274529244502887464, −5.47156165595483659599096975733, −5.18891765822095017483420416119, −4.01716907375447724638111147930, −3.78106439838390656310252233410, −2.91728336039430706781360995339, −2.82157880809136794235737109534, −1.48454647706044124862202797041, −1.23553727931513015429673319306, 1.23553727931513015429673319306, 1.48454647706044124862202797041, 2.82157880809136794235737109534, 2.91728336039430706781360995339, 3.78106439838390656310252233410, 4.01716907375447724638111147930, 5.18891765822095017483420416119, 5.47156165595483659599096975733, 5.85860392646274529244502887464, 6.11892232742314159672942746422, 7.14265581430412680312797508121, 7.45629129092727535273265018336, 7.63984826694279960957230548952, 8.412426757454182326040029176675, 8.835877260878964498105470781262, 9.142962783455452026301481432854, 9.688946276586426160804765139668, 9.731319955720249993534157848566, 10.72778583873254232817321505124, 10.73965459330473160920261018699

Graph of the ZZ-function along the critical line