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Properties

Label 8-252e4-1.1-c7e4-0-1
Degree 88
Conductor 40327580164032758016
Sign 11
Analytic cond. 3.84028×1073.84028\times 10^{7}
Root an. cond. 8.872488.87248
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.37e3·7-s + 1.32e4·13-s − 9.18e3·19-s − 2.39e4·25-s − 4.45e5·31-s − 8.52e5·37-s + 1.50e5·43-s + 1.17e6·49-s − 1.30e6·61-s − 6.07e6·67-s + 3.63e6·73-s − 1.68e7·79-s − 1.82e7·91-s − 3.64e7·97-s − 3.09e7·103-s − 3.19e7·109-s − 6.66e7·121-s + 127-s + 131-s + 1.26e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.67·13-s − 0.307·19-s − 0.307·25-s − 2.68·31-s − 2.76·37-s + 0.288·43-s + 10/7·49-s − 0.734·61-s − 2.46·67-s + 1.09·73-s − 3.85·79-s − 2.53·91-s − 4.05·97-s − 2.78·103-s − 2.36·109-s − 3.41·121-s + 0.464·133-s − 2.13·169-s + ⋯

Functional equation

Λ(s)=((28⋅38⋅74)s/2 ΓC(s)4 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}
Λ(s)=((28⋅38⋅74)s/2 ΓC(s+7/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅38⋅742^{8} \cdot 3^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 3.84028×1073.84028\times 10^{7}
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 28⋅38⋅74, ( :7/2,7/2,7/2,7/2), 1)(8,\ 2^{8} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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+p3T)4 ( 1 + p^{3} T )^{4}
good5C22≀C2C_2^2 \wr C_2 1+23988T2−337917994p2T4+23988p14T6+p28T8 1 + 23988 T^{2} - 337917994 p^{2} T^{4} + 23988 p^{14} T^{6} + p^{28} T^{8}
11C22≀C2C_2^2 \wr C_2 1+66638700T2+1869640076065382T4+66638700p14T6+p28T8 1 + 66638700 T^{2} + 1869640076065382 T^{4} + 66638700 p^{14} T^{6} + p^{28} T^{8}
13D4D_{4} (1−6636T+132998174T2−6636p7T3+p14T4)2 ( 1 - 6636 T + 132998174 T^{2} - 6636 p^{7} T^{3} + p^{14} T^{4} )^{2}
17C22≀C2C_2^2 \wr C_2 1−47340540T2+304481267672814902T4−47340540p14T6+p28T8 1 - 47340540 T^{2} + 304481267672814902 T^{4} - 47340540 p^{14} T^{6} + p^{28} T^{8}
19D4D_{4} (1+4592T+779207718T2+4592p7T3+p14T4)2 ( 1 + 4592 T + 779207718 T^{2} + 4592 p^{7} T^{3} + p^{14} T^{4} )^{2}
23C22≀C2C_2^2 \wr C_2 1−1458162852T2+22257185194263703190T4−1458162852p14T6+p28T8 1 - 1458162852 T^{2} + 22257185194263703190 T^{4} - 1458162852 p^{14} T^{6} + p^{28} T^{8}
29C22≀C2C_2^2 \wr C_2 1+38888338836T2+ 1 + 38888338836 T^{2} + 77 ⁣⋯ ⁣6277\!\cdots\!62T4+38888338836p14T6+p28T8 T^{4} + 38888338836 p^{14} T^{6} + p^{28} T^{8}
31D4D_{4} (1+222768T+37091070062T2+222768p7T3+p14T4)2 ( 1 + 222768 T + 37091070062 T^{2} + 222768 p^{7} T^{3} + p^{14} T^{4} )^{2}
37D4D_{4} (1+426404T+127886837070T2+426404p7T3+p14T4)2 ( 1 + 426404 T + 127886837070 T^{2} + 426404 p^{7} T^{3} + p^{14} T^{4} )^{2}
41C22≀C2C_2^2 \wr C_2 1+20538923492T2− 1 + 20538923492 T^{2} - 33 ⁣⋯ ⁣1833\!\cdots\!18T4+20538923492p14T6+p28T8 T^{4} + 20538923492 p^{14} T^{6} + p^{28} T^{8}
43D4D_{4} (1−75112T−159018745386T2−75112p7T3+p14T4)2 ( 1 - 75112 T - 159018745386 T^{2} - 75112 p^{7} T^{3} + p^{14} T^{4} )^{2}
47C22≀C2C_2^2 \wr C_2 1+1246374912572T2+ 1 + 1246374912572 T^{2} + 89 ⁣⋯ ⁣5089\!\cdots\!50T4+1246374912572p14T6+p28T8 T^{4} + 1246374912572 p^{14} T^{6} + p^{28} T^{8}
53C22≀C2C_2^2 \wr C_2 1−1583203178380T2+ 1 - 1583203178380 T^{2} + 33 ⁣⋯ ⁣4233\!\cdots\!42T4−1583203178380p14T6+p28T8 T^{4} - 1583203178380 p^{14} T^{6} + p^{28} T^{8}
59C22≀C2C_2^2 \wr C_2 1+4638308305356T2+ 1 + 4638308305356 T^{2} + 10 ⁣⋯ ⁣2210\!\cdots\!22T4+4638308305356p14T6+p28T8 T^{4} + 4638308305356 p^{14} T^{6} + p^{28} T^{8}
61D4D_{4} (1+651420T+6011864979998T2+651420p7T3+p14T4)2 ( 1 + 651420 T + 6011864979998 T^{2} + 651420 p^{7} T^{3} + p^{14} T^{4} )^{2}
67D4D_{4} (1+3035848T+13583249522022T2+3035848p7T3+p14T4)2 ( 1 + 3035848 T + 13583249522022 T^{2} + 3035848 p^{7} T^{3} + p^{14} T^{4} )^{2}
71C22≀C2C_2^2 \wr C_2 1+18965138502300T2+ 1 + 18965138502300 T^{2} + 18 ⁣⋯ ⁣6218\!\cdots\!62T4+18965138502300p14T6+p28T8 T^{4} + 18965138502300 p^{14} T^{6} + p^{28} T^{8}
73D4D_{4} (1−1815884T+3480307846614T2−1815884p7T3+p14T4)2 ( 1 - 1815884 T + 3480307846614 T^{2} - 1815884 p^{7} T^{3} + p^{14} T^{4} )^{2}
79D4D_{4} (1+8449344T+55594921645598T2+8449344p7T3+p14T4)2 ( 1 + 8449344 T + 55594921645598 T^{2} + 8449344 p^{7} T^{3} + p^{14} T^{4} )^{2}
83C22≀C2C_2^2 \wr C_2 1+42171068827884T2+ 1 + 42171068827884 T^{2} + 19 ⁣⋯ ⁣2219\!\cdots\!22T4+42171068827884p14T6+p28T8 T^{4} + 42171068827884 p^{14} T^{6} + p^{28} T^{8}
89C22≀C2C_2^2 \wr C_2 1+125209641009956T2+ 1 + 125209641009956 T^{2} + 78 ⁣⋯ ⁣2278\!\cdots\!22T4+125209641009956p14T6+p28T8 T^{4} + 125209641009956 p^{14} T^{6} + p^{28} T^{8}
97D4D_{4} (1+18246004T+241359719968806T2+18246004p7T3+p14T4)2 ( 1 + 18246004 T + 241359719968806 T^{2} + 18246004 p^{7} T^{3} + p^{14} T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.176804960700087156001750673924, −7.52746026032027695967384282822, −7.38897798401130006308341552390, −7.18871905068514839364563740478, −7.12210586512274957371632939359, −6.53580384491488422215690756300, −6.42302130216767427645239305831, −6.15116603126439566135620274415, −6.06556824906779349145964674417, −5.55121218188553383162781501693, −5.33744115820602921169840919180, −5.16340727838930836059552615389, −4.97208592264084290486570141198, −4.03165714156314589394822479088, −3.98932921317327353526885517996, −3.90149733788941743718702981773, −3.82574341543921192698948856484, −3.19372785401240981995850190577, −2.84828088185722445338837904774, −2.80851202304802577507559877586, −2.39203033339252285557403659578, −1.76715091672737984031257778312, −1.39743072023695782493108300958, −1.38767532699371223652205802179, −1.13582202255186179223433697523, 0, 0, 0, 0, 1.13582202255186179223433697523, 1.38767532699371223652205802179, 1.39743072023695782493108300958, 1.76715091672737984031257778312, 2.39203033339252285557403659578, 2.80851202304802577507559877586, 2.84828088185722445338837904774, 3.19372785401240981995850190577, 3.82574341543921192698948856484, 3.90149733788941743718702981773, 3.98932921317327353526885517996, 4.03165714156314589394822479088, 4.97208592264084290486570141198, 5.16340727838930836059552615389, 5.33744115820602921169840919180, 5.55121218188553383162781501693, 6.06556824906779349145964674417, 6.15116603126439566135620274415, 6.42302130216767427645239305831, 6.53580384491488422215690756300, 7.12210586512274957371632939359, 7.18871905068514839364563740478, 7.38897798401130006308341552390, 7.52746026032027695967384282822, 8.176804960700087156001750673924

Graph of the ZZ-function along the critical line