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Properties

Label 8-336e4-1.1-c3e4-0-4
Degree $8$
Conductor $12745506816$
Sign $1$
Analytic cond. $154461.$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 5·5-s + 9·9-s + 67·11-s + 82·13-s − 30·15-s + 92·17-s + 43·19-s + 148·23-s − 80·25-s − 54·27-s + 154·29-s − 520·31-s + 402·33-s − 7·37-s + 492·39-s − 852·41-s + 214·43-s − 45·45-s + 576·47-s + 659·49-s + 552·51-s + 243·53-s − 335·55-s + 258·57-s − 7·59-s − 224·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1/3·9-s + 1.83·11-s + 1.74·13-s − 0.516·15-s + 1.31·17-s + 0.519·19-s + 1.34·23-s − 0.639·25-s − 0.384·27-s + 0.986·29-s − 3.01·31-s + 2.12·33-s − 0.0311·37-s + 2.02·39-s − 3.24·41-s + 0.758·43-s − 0.149·45-s + 1.78·47-s + 1.92·49-s + 1.51·51-s + 0.629·53-s − 0.821·55-s + 0.599·57-s − 0.0154·59-s − 0.470·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(154461.\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(10.43922904\)
\(L(\frac12)\) \(\approx\) \(10.43922904\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 659 T^{2} + p^{6} T^{4} \)
good5$D_4\times C_2$ \( 1 + p T + 21 p T^{2} - 66 p^{2} T^{3} - 494 p^{2} T^{4} - 66 p^{5} T^{5} + 21 p^{7} T^{6} + p^{10} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 67 T + 1041 T^{2} - 52662 T^{3} + 4142284 T^{4} - 52662 p^{3} T^{5} + 1041 p^{6} T^{6} - 67 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 41 T + 4478 T^{2} - 41 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 92 T + 1902 T^{2} + 17664 p T^{3} - 78413 p^{2} T^{4} + 17664 p^{4} T^{5} + 1902 p^{6} T^{6} - 92 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 - 43 T - 9305 T^{2} + 110252 T^{3} + 64683544 T^{4} + 110252 p^{3} T^{5} - 9305 p^{6} T^{6} - 43 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 148 T - 2526 T^{2} - 14208 T^{3} + 182283043 T^{4} - 14208 p^{3} T^{5} - 2526 p^{6} T^{6} - 148 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 - 77 T + 9574 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 520 T + 144563 T^{2} + 34452600 T^{3} + 6891960488 T^{4} + 34452600 p^{3} T^{5} + 144563 p^{6} T^{6} + 520 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 + 7 T - 74033 T^{2} - 190568 T^{3} + 2919934318 T^{4} - 190568 p^{3} T^{5} - 74033 p^{6} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 426 T + 171106 T^{2} + 426 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 107 T + 86220 T^{2} - 107 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 576 T + 89606 T^{2} - 19885824 T^{3} + 13421113923 T^{4} - 19885824 p^{3} T^{5} + 89606 p^{6} T^{6} - 576 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 243 T - 250441 T^{2} - 2851848 T^{3} + 64828660998 T^{4} - 2851848 p^{3} T^{5} - 250441 p^{6} T^{6} - 243 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 + 7 T - 200565 T^{2} - 1471008 T^{3} - 1944620216 T^{4} - 1471008 p^{3} T^{5} - 200565 p^{6} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 224 T - 410950 T^{2} + 1604736 T^{3} + 149727814859 T^{4} + 1604736 p^{3} T^{5} - 410950 p^{6} T^{6} + 224 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 687 T - 206863 T^{2} - 53109222 T^{3} + 228403689708 T^{4} - 53109222 p^{3} T^{5} - 206863 p^{6} T^{6} - 687 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 921 T + 270053 T^{2} + 184058166 T^{3} - 147013032042 T^{4} + 184058166 p^{3} T^{5} + 270053 p^{6} T^{6} - 921 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 526 T - 757051 T^{2} + 25063374 T^{3} + 689091996644 T^{4} + 25063374 p^{3} T^{5} - 757051 p^{6} T^{6} + 526 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 221 T + 945628 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 774 T - 367486 T^{2} - 343173024 T^{3} + 14930800239 T^{4} - 343173024 p^{3} T^{5} - 367486 p^{6} T^{6} + 774 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 1953 T + 2366992 T^{2} - 1953 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.946152942935418293838386317322, −7.78352013800705380088756847472, −7.21973000688761068561840026897, −7.13501768405364311196247999584, −7.11446586397101746615383553202, −6.95635872653857160564095626899, −6.34732734679273777043322519657, −6.09308922841696520657010820932, −5.80814288417660007423397918923, −5.66357500489084533147386898518, −5.43800432911910498537792270691, −4.88767650622385544488445734552, −4.77683645063327750899184015223, −4.11697109124252165276177600566, −4.04610813224077190537437258688, −3.51354637359241987015262429787, −3.50910206829972718130434729273, −3.49570239090395751747798361021, −2.97229362358580741131616491906, −2.47473992735346134983853319325, −2.01258290574865809371375266234, −1.62407672577037477706632327052, −1.31940056525079763941896465414, −0.877850252177957969167565886881, −0.51553774487952311241337788124, 0.51553774487952311241337788124, 0.877850252177957969167565886881, 1.31940056525079763941896465414, 1.62407672577037477706632327052, 2.01258290574865809371375266234, 2.47473992735346134983853319325, 2.97229362358580741131616491906, 3.49570239090395751747798361021, 3.50910206829972718130434729273, 3.51354637359241987015262429787, 4.04610813224077190537437258688, 4.11697109124252165276177600566, 4.77683645063327750899184015223, 4.88767650622385544488445734552, 5.43800432911910498537792270691, 5.66357500489084533147386898518, 5.80814288417660007423397918923, 6.09308922841696520657010820932, 6.34732734679273777043322519657, 6.95635872653857160564095626899, 7.11446586397101746615383553202, 7.13501768405364311196247999584, 7.21973000688761068561840026897, 7.78352013800705380088756847472, 7.946152942935418293838386317322

Graph of the $Z$-function along the critical line