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Properties

Label 4-9360e2-1.1-c1e2-0-20
Degree 44
Conductor 8760960087609600
Sign 11
Analytic cond. 5586.065586.06
Root an. cond. 8.645228.64522
Motivic weight 11
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  + 2·5-s + 5·7-s − 11-s − 2·13-s − 17-s + 2·19-s − 9·23-s + 3·25-s − 6·29-s − 12·31-s + 10·35-s − 9·37-s + 3·41-s + 2·43-s − 14·47-s + 9·49-s + 3·53-s − 2·55-s − 24·59-s − 61-s − 4·65-s − 2·67-s − 25·71-s − 12·73-s − 5·77-s + 3·79-s − 2·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.88·7-s − 0.301·11-s − 0.554·13-s − 0.242·17-s + 0.458·19-s − 1.87·23-s + 3/5·25-s − 1.11·29-s − 2.15·31-s + 1.69·35-s − 1.47·37-s + 0.468·41-s + 0.304·43-s − 2.04·47-s + 9/7·49-s + 0.412·53-s − 0.269·55-s − 3.12·59-s − 0.128·61-s − 0.496·65-s − 0.244·67-s − 2.96·71-s − 1.40·73-s − 0.569·77-s + 0.337·79-s − 0.216·85-s + ⋯

Functional equation

Λ(s)=(87609600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(87609600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8760960087609600    =    28⋅34⋅52⋅1322^{8} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 5586.065586.06
Root analytic conductor: 8.645228.64522
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 87609600, ( :1/2,1/2), 1)(4,\ 87609600,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
13C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1−5T+16T2−5pT3+p2T4 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4}
11D4D_{4} 1+T+18T2+pT3+p2T4 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4}
17D4D_{4} 1+T+30T2+pT3+p2T4 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4}
19D4D_{4} 1−2T+22T2−2pT3+p2T4 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+9T+62T2+9pT3+p2T4 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+6T+50T2+6pT3+p2T4 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4}
31C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
37D4D_{4} 1+9T+56T2+9pT3+p2T4 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−3T+80T2−3pT3+p2T4 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−2T+70T2−2pT3+p2T4 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+14T+126T2+14pT3+p2T4 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−3T+70T2−3pT3+p2T4 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4}
59C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
61D4D_{4} 1+T+84T2+pT3+p2T4 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4}
67D4D_{4} 1+2T−18T2+2pT3+p2T4 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+25T+294T2+25pT3+p2T4 1 + 25 T + 294 T^{2} + 25 p T^{3} + p^{2} T^{4}
73C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
79D4D_{4} 1−3T+122T2−3pT3+p2T4 1 - 3 T + 122 T^{2} - 3 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
89D4D_{4} 1−9T+160T2−9pT3+p2T4 1 - 9 T + 160 T^{2} - 9 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−5T−8T2−5pT3+p2T4 1 - 5 T - 8 T^{2} - 5 p T^{3} + p^{2} 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

−7.63745624210096357171202597080, −7.53118532001094719175023556090, −6.70668270789967633890177593551, −6.63264688609698015462378773702, −6.00866554670626865042079744714, −5.79986410952386088974612850144, −5.37785113468469007359117471042, −5.25177300777133945870242579394, −4.77639799981178319614907430538, −4.49960807842746904731067276608, −4.16076445854621581259954130651, −3.61881409878745915715574603080, −3.22703814353017751886902231326, −2.79710430694435538976692478721, −2.07720068936321360482182710078, −1.97255672866968911267077175713, −1.48507205216074216520246176703, −1.43757561007400231991021636840, 0, 0, 1.43757561007400231991021636840, 1.48507205216074216520246176703, 1.97255672866968911267077175713, 2.07720068936321360482182710078, 2.79710430694435538976692478721, 3.22703814353017751886902231326, 3.61881409878745915715574603080, 4.16076445854621581259954130651, 4.49960807842746904731067276608, 4.77639799981178319614907430538, 5.25177300777133945870242579394, 5.37785113468469007359117471042, 5.79986410952386088974612850144, 6.00866554670626865042079744714, 6.63264688609698015462378773702, 6.70668270789967633890177593551, 7.53118532001094719175023556090, 7.63745624210096357171202597080

Graph of the ZZ-function along the critical line