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Properties

Label 4-588e2-1.1-c1e2-0-14
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 6·11-s − 4·13-s − 4·19-s + 6·23-s + 5·25-s − 27-s + 12·29-s + 8·31-s + 6·33-s − 2·37-s − 4·39-s − 24·41-s − 8·43-s + 12·47-s + 6·53-s − 4·57-s − 10·61-s − 8·67-s + 6·69-s + 12·71-s − 10·73-s + 5·75-s + 4·79-s − 81-s + 24·83-s + 12·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.80·11-s − 1.10·13-s − 0.917·19-s + 1.25·23-s + 25-s − 0.192·27-s + 2.22·29-s + 1.43·31-s + 1.04·33-s − 0.328·37-s − 0.640·39-s − 3.74·41-s − 1.21·43-s + 1.75·47-s + 0.824·53-s − 0.529·57-s − 1.28·61-s − 0.977·67-s + 0.722·69-s + 1.42·71-s − 1.17·73-s + 0.577·75-s + 0.450·79-s − 1/9·81-s + 2.63·83-s + 1.28·87-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 2.4477933012.447793301
L(12)L(\frac12) ≈\approx 2.4477933012.447793301
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−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
11C22C_2^2 1−6T+25T2−6pT3+p2T4 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
19C22C_2^2 1+4T−3T2+4pT3+p2T4 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−6T+13T2−6pT3+p2T4 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C22C_2^2 1−8T+33T2−8pT3+p2T4 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+2T−33T2+2pT3+p2T4 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4}
41C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1−12T+97T2−12pT3+p2T4 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−6T−17T2−6pT3+p2T4 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C22C_2^2 1+10T+39T2+10pT3+p2T4 1 + 10 T + 39 T^{2} + 10 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 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C2C_2 (1−7T+pT2)(1+17T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} )
79C2C_2 (1−17T+pT2)(1+13T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} )
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C22C_2^2 1−12T+55T2−12pT3+p2T4 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4}
97C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
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   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

−10.59523025583029202558354503309, −10.47943702209173908529826735682, −10.16106844341492498816097585141, −9.458419761601227879363089501348, −9.116124051667141690885045631184, −8.702602132672200820601415364440, −8.443935749541386460275208542013, −7.978727741737425187783778722772, −7.19367553515283518122311309949, −6.80379843620544208445214810698, −6.55876832554089240168348208687, −6.18573546364636573254799723821, −5.02499277950116262149259163091, −4.94150342410443692419104271924, −4.40520587979508663278659154208, −3.63324610500658274442201888781, −3.15071750733572713514677841518, −2.58247034454246733586331312893, −1.77118414331097806789375930088, −0.914534402024743180882876203600, 0.914534402024743180882876203600, 1.77118414331097806789375930088, 2.58247034454246733586331312893, 3.15071750733572713514677841518, 3.63324610500658274442201888781, 4.40520587979508663278659154208, 4.94150342410443692419104271924, 5.02499277950116262149259163091, 6.18573546364636573254799723821, 6.55876832554089240168348208687, 6.80379843620544208445214810698, 7.19367553515283518122311309949, 7.978727741737425187783778722772, 8.443935749541386460275208542013, 8.702602132672200820601415364440, 9.116124051667141690885045631184, 9.458419761601227879363089501348, 10.16106844341492498816097585141, 10.47943702209173908529826735682, 10.59523025583029202558354503309

Graph of the ZZ-function along the critical line