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Properties

Label 4-192e2-1.1-c4e2-0-1
Degree 44
Conductor 3686436864
Sign 11
Analytic cond. 393.904393.904
Root an. cond. 4.455004.45500
Motivic weight 44
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  − 12·5-s − 27·9-s + 172·13-s + 852·17-s − 1.14e3·25-s − 2.36e3·29-s + 860·37-s + 4.50e3·41-s + 324·45-s + 914·49-s + 3.20e3·53-s − 4.22e3·61-s − 2.06e3·65-s + 8.13e3·73-s + 729·81-s − 1.02e4·85-s − 4.09e3·89-s − 5.88e3·97-s + 2.33e4·101-s − 3.50e4·109-s + 2.40e4·113-s − 4.64e3·117-s − 5.71e3·121-s + 2.16e4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.479·5-s − 1/3·9-s + 1.01·13-s + 2.94·17-s − 1.82·25-s − 2.81·29-s + 0.628·37-s + 2.67·41-s + 4/25·45-s + 0.380·49-s + 1.14·53-s − 1.13·61-s − 0.488·65-s + 1.52·73-s + 1/9·81-s − 1.41·85-s − 0.516·89-s − 0.625·97-s + 2.29·101-s − 2.95·109-s + 1.88·113-s − 0.339·117-s − 0.390·121-s + 1.38·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯

Functional equation

Λ(s)=(36864s/2 ΓC(s)2 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(36864s/2 ΓC(s+2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3686436864    =    212⋅322^{12} \cdot 3^{2}
Sign: 11
Analytic conductor: 393.904393.904
Root analytic conductor: 4.455004.45500
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 36864, ( :2,2), 1)(4,\ 36864,\ (\ :2, 2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.2560793412.256079341
L(12)L(\frac12) ≈\approx 2.2560793412.256079341
L(3)L(3) 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+p3T2 1 + p^{3} T^{2}
good5C2C_2 (1+6T+p4T2)2 ( 1 + 6 T + p^{4} T^{2} )^{2}
7C22C_2^2 1−914T2+p8T4 1 - 914 T^{2} + p^{8} T^{4}
11C22C_2^2 1+5710T2+p8T4 1 + 5710 T^{2} + p^{8} T^{4}
13C2C_2 (1−86T+p4T2)2 ( 1 - 86 T + p^{4} T^{2} )^{2}
17C2C_2 (1−426T+p4T2)2 ( 1 - 426 T + p^{4} T^{2} )^{2}
19C22C_2^2 1−256754T2+p8T4 1 - 256754 T^{2} + p^{8} T^{4}
23C22C_2^2 1+190T2+p8T4 1 + 190 T^{2} + p^{8} T^{4}
29C2C_2 (1+1182T+p4T2)2 ( 1 + 1182 T + p^{4} T^{2} )^{2}
31C22C_2^2 1+582958T2+p8T4 1 + 582958 T^{2} + p^{8} T^{4}
37C2C_2 (1−430T+p4T2)2 ( 1 - 430 T + p^{4} T^{2} )^{2}
41C2C_2 (1−2250T+p4T2)2 ( 1 - 2250 T + p^{4} T^{2} )^{2}
43C22C_2^2 1+190p2T2+p8T4 1 + 190 p^{2} T^{2} + p^{8} T^{4}
47C22C_2^2 1−9619394T2+p8T4 1 - 9619394 T^{2} + p^{8} T^{4}
53C2C_2 (1−1602T+p4T2)2 ( 1 - 1602 T + p^{4} T^{2} )^{2}
59C22C_2^2 1−11602610T2+p8T4 1 - 11602610 T^{2} + p^{8} T^{4}
61C2C_2 (1+2114T+p4T2)2 ( 1 + 2114 T + p^{4} T^{2} )^{2}
67C22C_2^2 1−38587634T2+p8T4 1 - 38587634 T^{2} + p^{8} T^{4}
71C22C_2^2 1−41865410T2+p8T4 1 - 41865410 T^{2} + p^{8} T^{4}
73C2C_2 (1−4066T+p4T2)2 ( 1 - 4066 T + p^{4} T^{2} )^{2}
79C22C_2^2 1−47103314T2+p8T4 1 - 47103314 T^{2} + p^{8} T^{4}
83C22C_2^2 1−10900850T2+p8T4 1 - 10900850 T^{2} + p^{8} T^{4}
89C2C_2 (1+2046T+p4T2)2 ( 1 + 2046 T + p^{4} T^{2} )^{2}
97C2C_2 (1+2942T+p4T2)2 ( 1 + 2942 T + p^{4} 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

−12.03988071947171974455410682871, −11.69999369681912621665587196364, −10.99839995521059931415841775055, −10.91426823494330350015846874727, −10.05983995032866721134367769504, −9.452945944375264307741247581544, −9.407196776574690830858750515600, −8.480996090906597778643611105520, −7.80808189777775110640886760262, −7.71901308396404731537621710080, −7.23076540704023610713578176636, −6.08063625989352125452673947908, −5.62886533152600222385891439528, −5.62176299669672406973026601249, −4.34595465206099892134215743540, −3.63621772811811975782906290114, −3.48608396289622738793019560478, −2.35189459784699262891500270857, −1.38604910845612020864876934199, −0.58540537027730307109301864504, 0.58540537027730307109301864504, 1.38604910845612020864876934199, 2.35189459784699262891500270857, 3.48608396289622738793019560478, 3.63621772811811975782906290114, 4.34595465206099892134215743540, 5.62176299669672406973026601249, 5.62886533152600222385891439528, 6.08063625989352125452673947908, 7.23076540704023610713578176636, 7.71901308396404731537621710080, 7.80808189777775110640886760262, 8.480996090906597778643611105520, 9.407196776574690830858750515600, 9.452945944375264307741247581544, 10.05983995032866721134367769504, 10.91426823494330350015846874727, 10.99839995521059931415841775055, 11.69999369681912621665587196364, 12.03988071947171974455410682871

Graph of the ZZ-function along the critical line