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Properties

Label 8-525e4-1.1-c3e4-0-5
Degree 88
Conductor 7596914062575969140625
Sign 11
Analytic cond. 920664.920664.
Root an. cond. 5.565605.56560
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·4-s − 18·9-s − 184·11-s + 99·16-s + 216·19-s + 472·29-s − 120·31-s − 252·36-s + 88·41-s − 2.57e3·44-s − 98·49-s + 928·59-s − 1.36e3·61-s + 924·64-s − 1.48e3·71-s + 3.02e3·76-s + 816·79-s + 243·81-s + 2.66e3·89-s + 3.31e3·99-s + 136·101-s − 5.27e3·109-s + 6.60e3·116-s + 1.58e4·121-s − 1.68e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 7/4·4-s − 2/3·9-s − 5.04·11-s + 1.54·16-s + 2.60·19-s + 3.02·29-s − 0.695·31-s − 7/6·36-s + 0.335·41-s − 8.82·44-s − 2/7·49-s + 2.04·59-s − 2.87·61-s + 1.80·64-s − 2.47·71-s + 4.56·76-s + 1.16·79-s + 1/3·81-s + 3.17·89-s + 3.36·99-s + 0.133·101-s − 4.63·109-s + 5.28·116-s + 11.9·121-s − 1.21·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((34⋅58⋅74)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅58⋅743^{4} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 920664.920664.
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅74, ( :3/2,3/2,3/2,3/2), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 3.1421039833.142103983
L(12)L(\frac12) ≈\approx 3.1421039833.142103983
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)
bad3C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
5 1 1
7C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
good2D4×C2D_4\times C_2 1−7pT2+97T4−7p7T6+p12T8 1 - 7 p T^{2} + 97 T^{4} - 7 p^{7} T^{6} + p^{12} T^{8}
11D4D_{4} (1+92T+4758T2+92p3T3+p6T4)2 ( 1 + 92 T + 4758 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} )^{2}
13D4×C2D_4\times C_2 1+5684T2+17268502T4+5684p6T6+p12T8 1 + 5684 T^{2} + 17268502 T^{4} + 5684 p^{6} T^{6} + p^{12} T^{8}
17D4×C2D_4\times C_2 1+676T2+29648902T4+676p6T6+p12T8 1 + 676 T^{2} + 29648902 T^{4} + 676 p^{6} T^{6} + p^{12} T^{8}
19D4D_{4} (1−108T+13254T2−108p3T3+p6T4)2 ( 1 - 108 T + 13254 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} )^{2}
23D4×C2D_4\times C_2 1+284pT2+101938534T4+284p7T6+p12T8 1 + 284 p T^{2} + 101938534 T^{4} + 284 p^{7} T^{6} + p^{12} T^{8}
29D4D_{4} (1−236T+61982T2−236p3T3+p6T4)2 ( 1 - 236 T + 61982 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} )^{2}
31D4D_{4} (1+60T+8462T2+60p3T3+p6T4)2 ( 1 + 60 T + 8462 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{2}
37D4×C2D_4\times C_2 1−176044T2+12759471222T4−176044p6T6+p12T8 1 - 176044 T^{2} + 12759471222 T^{4} - 176044 p^{6} T^{6} + p^{12} T^{8}
41D4D_{4} (1−44T+106326T2−44p3T3+p6T4)2 ( 1 - 44 T + 106326 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} )^{2}
43D4×C2D_4\times C_2 1−144940T2+16379434678T4−144940p6T6+p12T8 1 - 144940 T^{2} + 16379434678 T^{4} - 144940 p^{6} T^{6} + p^{12} T^{8}
47D4×C2D_4\times C_2 1−53052T2−317140666T4−53052p6T6+p12T8 1 - 53052 T^{2} - 317140666 T^{4} - 53052 p^{6} T^{6} + p^{12} T^{8}
53D4×C2D_4\times C_2 1−521420T2+112288959478T4−521420p6T6+p12T8 1 - 521420 T^{2} + 112288959478 T^{4} - 521420 p^{6} T^{6} + p^{12} T^{8}
59D4D_{4} (1−464T+458102T2−464p3T3+p6T4)2 ( 1 - 464 T + 458102 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} )^{2}
61D4D_{4} (1+684T+535646T2+684p3T3+p6T4)2 ( 1 + 684 T + 535646 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} )^{2}
67D4×C2D_4\times C_2 1−663244T2+218042637142T4−663244p6T6+p12T8 1 - 663244 T^{2} + 218042637142 T^{4} - 663244 p^{6} T^{6} + p^{12} T^{8}
71D4D_{4} (1+740T+757502T2+740p3T3+p6T4)2 ( 1 + 740 T + 757502 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} )^{2}
73D4×C2D_4\times C_2 1−1317340T2+723135691558T4−1317340p6T6+p12T8 1 - 1317340 T^{2} + 723135691558 T^{4} - 1317340 p^{6} T^{6} + p^{12} T^{8}
79D4D_{4} (1−408T−143586T2−408p3T3+p6T4)2 ( 1 - 408 T - 143586 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} )^{2}
83D4×C2D_4\times C_2 1−2031756T2+1672847111702T4−2031756p6T6+p12T8 1 - 2031756 T^{2} + 1672847111702 T^{4} - 2031756 p^{6} T^{6} + p^{12} T^{8}
89D4D_{4} (1−1332T+1790774T2−1332p3T3+p6T4)2 ( 1 - 1332 T + 1790774 T^{2} - 1332 p^{3} T^{3} + p^{6} T^{4} )^{2}
97D4×C2D_4\times C_2 1−580380T2+1528544052038T4−580380p6T6+p12T8 1 - 580380 T^{2} + 1528544052038 T^{4} - 580380 p^{6} T^{6} + p^{12} T^{8}
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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

−7.50474014685843443155313603229, −7.38996629216998087385113079209, −6.98237075153736080608079536876, −6.59731488510146135835771490082, −6.53484511589033305771407784021, −6.23086564069295230248777149662, −5.70485548349316412714677842163, −5.68003048887681946222963262919, −5.57058158395315353343262965099, −5.15256557484956982259026126728, −4.90853104921410795360070234082, −4.83976750361355594094720533691, −4.75149722176922339173200229711, −3.88900486469867854799395718820, −3.68159579613682151539571364965, −3.13253116104481272972588001508, −2.95761621223385487129699275954, −2.83528033427096463647666197159, −2.62745804493684440791573587046, −2.37999062387175313573331307629, −2.18498110468207537380224385567, −1.44632169346843767737681811884, −1.21100416868405288663222589763, −0.54152657936631692433730521784, −0.30721832406503930222481964623, 0.30721832406503930222481964623, 0.54152657936631692433730521784, 1.21100416868405288663222589763, 1.44632169346843767737681811884, 2.18498110468207537380224385567, 2.37999062387175313573331307629, 2.62745804493684440791573587046, 2.83528033427096463647666197159, 2.95761621223385487129699275954, 3.13253116104481272972588001508, 3.68159579613682151539571364965, 3.88900486469867854799395718820, 4.75149722176922339173200229711, 4.83976750361355594094720533691, 4.90853104921410795360070234082, 5.15256557484956982259026126728, 5.57058158395315353343262965099, 5.68003048887681946222963262919, 5.70485548349316412714677842163, 6.23086564069295230248777149662, 6.53484511589033305771407784021, 6.59731488510146135835771490082, 6.98237075153736080608079536876, 7.38996629216998087385113079209, 7.50474014685843443155313603229

Graph of the ZZ-function along the critical line