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Properties

Label 4-9576e2-1.1-c1e2-0-6
Degree 44
Conductor 9169977691699776
Sign 11
Analytic cond. 5846.855846.85
Root an. cond. 8.744418.74441
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  + 2·5-s − 2·7-s + 8·13-s + 10·17-s + 2·19-s − 8·23-s − 4·25-s − 2·29-s − 4·35-s − 8·37-s + 16·41-s − 4·43-s − 6·47-s + 3·49-s + 6·53-s + 16·59-s + 16·61-s + 16·65-s − 8·67-s − 2·71-s + 8·73-s − 4·79-s + 10·83-s + 20·85-s − 16·91-s + 4·95-s + 12·97-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s + 2.21·13-s + 2.42·17-s + 0.458·19-s − 1.66·23-s − 4/5·25-s − 0.371·29-s − 0.676·35-s − 1.31·37-s + 2.49·41-s − 0.609·43-s − 0.875·47-s + 3/7·49-s + 0.824·53-s + 2.08·59-s + 2.04·61-s + 1.98·65-s − 0.977·67-s − 0.237·71-s + 0.936·73-s − 0.450·79-s + 1.09·83-s + 2.16·85-s − 1.67·91-s + 0.410·95-s + 1.21·97-s + ⋯

Functional equation

Λ(s)=(91699776s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(91699776s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9169977691699776    =    26⋅34⋅72⋅1922^{6} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 5846.855846.85
Root analytic conductor: 8.744418.74441
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 91699776, ( :1/2,1/2), 1)(4,\ 91699776,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.8530214974.853021497
L(12)L(\frac12) ≈\approx 4.8530214974.853021497
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
7C1C_1 (1+T)2 ( 1 + T )^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good5D4D_{4} 1−2T+8T2−2pT3+p2T4 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
17D4D_{4} 1−10T+56T2−10pT3+p2T4 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+8T+50T2+8pT3+p2T4 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+2T+56T2+2pT3+p2T4 1 + 2 T + 56 T^{2} + 2 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
37D4D_{4} 1+8T+78T2+8pT3+p2T4 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−16T+134T2−16pT3+p2T4 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+4T+42T2+4pT3+p2T4 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+6T+28T2+6pT3+p2T4 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−6T+88T2−6pT3+p2T4 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
61D4D_{4} 1−16T+174T2−16pT3+p2T4 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4}
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71D4D_{4} 1+2T+116T2+2pT3+p2T4 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−8T+150T2−8pT3+p2T4 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+4T+150T2+4pT3+p2T4 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−10T+164T2−10pT3+p2T4 1 - 10 T + 164 T^{2} - 10 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4}
97C2C_2 (1−6T+pT2)2 ( 1 - 6 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

−7.82508858757232868766556216147, −7.48966471687452736005384545564, −7.30300186993056739819950634438, −6.72033452185198052128553924444, −6.28104305201082942431105533896, −6.16354047907991182134082748996, −5.78211797709902953350540770050, −5.69475418493493397485105148642, −5.15437858554929948956192632811, −5.06474179561794531607418513894, −3.97313550308581969774860911952, −3.96967273762296694840005577259, −3.55399166021064744249868098288, −3.55163173685440234926923557660, −2.68504311789875791285292515414, −2.53939371161969454489779208379, −1.71977579391393835244646285291, −1.62569836511552949477421568566, −0.950352055993476671989546866435, −0.57458679088412248307946885761, 0.57458679088412248307946885761, 0.950352055993476671989546866435, 1.62569836511552949477421568566, 1.71977579391393835244646285291, 2.53939371161969454489779208379, 2.68504311789875791285292515414, 3.55163173685440234926923557660, 3.55399166021064744249868098288, 3.96967273762296694840005577259, 3.97313550308581969774860911952, 5.06474179561794531607418513894, 5.15437858554929948956192632811, 5.69475418493493397485105148642, 5.78211797709902953350540770050, 6.16354047907991182134082748996, 6.28104305201082942431105533896, 6.72033452185198052128553924444, 7.30300186993056739819950634438, 7.48966471687452736005384545564, 7.82508858757232868766556216147

Graph of the ZZ-function along the critical line