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Properties

Label 4-2496e2-1.1-c3e2-0-15
Degree 44
Conductor 62300166230016
Sign 11
Analytic cond. 21688.021688.0
Root an. cond. 12.135412.1354
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 6·5-s − 10·7-s + 27·9-s + 16·11-s − 26·13-s − 36·15-s + 4·17-s + 70·19-s − 60·21-s − 128·23-s − 110·25-s + 108·27-s + 80·29-s − 250·31-s + 96·33-s + 60·35-s + 152·37-s − 156·39-s − 146·41-s + 504·43-s − 162·45-s − 524·47-s − 498·49-s + 24·51-s + 52·53-s − 96·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.536·5-s − 0.539·7-s + 9-s + 0.438·11-s − 0.554·13-s − 0.619·15-s + 0.0570·17-s + 0.845·19-s − 0.623·21-s − 1.16·23-s − 0.879·25-s + 0.769·27-s + 0.512·29-s − 1.44·31-s + 0.506·33-s + 0.289·35-s + 0.675·37-s − 0.640·39-s − 0.556·41-s + 1.78·43-s − 0.536·45-s − 1.62·47-s − 1.45·49-s + 0.0658·51-s + 0.134·53-s − 0.235·55-s + ⋯

Functional equation

Λ(s)=(6230016s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(6230016s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 62300166230016    =    212⋅32⋅1322^{12} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 21688.021688.0
Root analytic conductor: 12.135412.1354
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 6230016, ( :3/2,3/2), 1)(4,\ 6230016,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
13C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1+6T+146T2+6p3T3+p6T4 1 + 6 T + 146 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+10T+598T2+10p3T3+p6T4 1 + 10 T + 598 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−16T+918T2−16p3T3+p6T4 1 - 16 T + 918 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}
17C2C_2 (1−2T+p3T2)2 ( 1 - 2 T + p^{3} T^{2} )^{2}
19D4D_{4} 1−70T+12118T2−70p3T3+p6T4 1 - 70 T + 12118 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4}
23C2C_2 (1+64T+p3T2)2 ( 1 + 64 T + p^{3} T^{2} )^{2}
29D4D_{4} 1−80T+46310T2−80p3T3+p6T4 1 - 80 T + 46310 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+250T+49782T2+250p3T3+p6T4 1 + 250 T + 49782 T^{2} + 250 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−152T+70470T2−152p3T3+p6T4 1 - 152 T + 70470 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+146T+137634T2+146p3T3+p6T4 1 + 146 T + 137634 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−504T+215286T2−504p3T3+p6T4 1 - 504 T + 215286 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+524T+272222T2+524p3T3+p6T4 1 + 524 T + 272222 T^{2} + 524 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−52T+291198T2−52p3T3+p6T4 1 - 52 T + 291198 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−164T+406182T2−164p3T3+p6T4 1 - 164 T + 406182 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−304T+237958T2−304p3T3+p6T4 1 - 304 T + 237958 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−914T+804838T2−914p3T3+p6T4 1 - 914 T + 804838 T^{2} - 914 p^{3} T^{3} + p^{6} T^{4}
71C22C_2^2 1+455470T2+p6T4 1 + 455470 T^{2} + p^{6} T^{4}
73D4D_{4} 1+456T+825950T2+456p3T3+p6T4 1 + 456 T + 825950 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+824T+1009374T2+824p3T3+p6T4 1 + 824 T + 1009374 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+828T+1032470T2+828p3T3+p6T4 1 + 828 T + 1032470 T^{2} + 828 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−826T+1571354T2−826p3T3+p6T4 1 - 826 T + 1571354 T^{2} - 826 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−552T+219630T2−552p3T3+p6T4 1 - 552 T + 219630 T^{2} - 552 p^{3} T^{3} + p^{6} T^{4}
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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

−8.237531694687323877835094608507, −8.015477672970482303670989439905, −7.65297196239295879219717400119, −7.47341296325385385718878937316, −6.75033180918405066368917242480, −6.73299933138166591858171015248, −6.11863476347586999451652403537, −5.64047112512998964043447999523, −5.22388813953890825643913289344, −4.73143305510029877797670287036, −4.10994378848860783003291260456, −3.94628964754630066782371273748, −3.47843182284336002331919620494, −3.14572912476690580172044778577, −2.45874610749956357644549805760, −2.26462285786915916032528024540, −1.47804494756137105180180355013, −1.12350226157738192658699375204, 0, 0, 1.12350226157738192658699375204, 1.47804494756137105180180355013, 2.26462285786915916032528024540, 2.45874610749956357644549805760, 3.14572912476690580172044778577, 3.47843182284336002331919620494, 3.94628964754630066782371273748, 4.10994378848860783003291260456, 4.73143305510029877797670287036, 5.22388813953890825643913289344, 5.64047112512998964043447999523, 6.11863476347586999451652403537, 6.73299933138166591858171015248, 6.75033180918405066368917242480, 7.47341296325385385718878937316, 7.65297196239295879219717400119, 8.015477672970482303670989439905, 8.237531694687323877835094608507

Graph of the ZZ-function along the critical line