Properties

Label 8-1232e4-1.1-c3e4-0-3
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $2.79194\times 10^{7}$
Root an. cond. $8.52586$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 19·5-s + 28·7-s − 31·9-s − 44·11-s + 2·13-s − 57·15-s + 8·17-s + 6·19-s + 84·21-s − 159·23-s − 129·25-s − 90·27-s + 144·29-s + 183·31-s − 132·33-s − 532·35-s − 475·37-s + 6·39-s − 768·41-s + 152·43-s + 589·45-s + 228·47-s + 490·49-s + 24·51-s + 396·53-s + 836·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.69·5-s + 1.51·7-s − 1.14·9-s − 1.20·11-s + 0.0426·13-s − 0.981·15-s + 0.114·17-s + 0.0724·19-s + 0.872·21-s − 1.44·23-s − 1.03·25-s − 0.641·27-s + 0.922·29-s + 1.06·31-s − 0.696·33-s − 2.56·35-s − 2.11·37-s + 0.0246·39-s − 2.92·41-s + 0.539·43-s + 1.95·45-s + 0.707·47-s + 10/7·49-s + 0.0658·51-s + 1.02·53-s + 2.04·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2.79194\times 10^{7}\)
Root analytic conductor: \(8.52586\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - p T )^{4} \)
11$C_1$ \( ( 1 + p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - p T + 40 T^{2} - 41 p T^{3} + 1318 T^{4} - 41 p^{4} T^{5} + 40 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 19 T + 98 p T^{2} + 1253 p T^{3} + 92394 T^{4} + 1253 p^{4} T^{5} + 98 p^{7} T^{6} + 19 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 2 T + 5904 T^{2} - 68750 T^{3} + 16352174 T^{4} - 68750 p^{3} T^{5} + 5904 p^{6} T^{6} - 2 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 8 T + 6692 T^{2} - 16832 T^{3} + 55319398 T^{4} - 16832 p^{3} T^{5} + 6692 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 6 T + 18156 T^{2} + 52826 T^{3} + 163953750 T^{4} + 52826 p^{3} T^{5} + 18156 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 159 T + 46268 T^{2} + 4706667 T^{3} + 802585638 T^{4} + 4706667 p^{3} T^{5} + 46268 p^{6} T^{6} + 159 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 144 T + 59772 T^{2} - 10743984 T^{3} + 1758200566 T^{4} - 10743984 p^{3} T^{5} + 59772 p^{6} T^{6} - 144 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 183 T + 108120 T^{2} - 16542811 T^{3} + 4666030854 T^{4} - 16542811 p^{3} T^{5} + 108120 p^{6} T^{6} - 183 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 475 T + 244318 T^{2} + 66539857 T^{3} + 18947050978 T^{4} + 66539857 p^{3} T^{5} + 244318 p^{6} T^{6} + 475 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 768 T + 455076 T^{2} + 171817752 T^{3} + 53269063174 T^{4} + 171817752 p^{3} T^{5} + 455076 p^{6} T^{6} + 768 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 152 T + 192060 T^{2} - 12011352 T^{3} + 17735402518 T^{4} - 12011352 p^{3} T^{5} + 192060 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 228 T + 195076 T^{2} - 44872492 T^{3} + 29572167094 T^{4} - 44872492 p^{3} T^{5} + 195076 p^{6} T^{6} - 228 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 396 T + 536244 T^{2} - 142154340 T^{3} + 112824313190 T^{4} - 142154340 p^{3} T^{5} + 536244 p^{6} T^{6} - 396 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 733 T + 366664 T^{2} - 158030589 T^{3} + 95193022454 T^{4} - 158030589 p^{3} T^{5} + 366664 p^{6} T^{6} - 733 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 1012 T + 1240676 T^{2} + 727427796 T^{3} + 460951794806 T^{4} + 727427796 p^{3} T^{5} + 1240676 p^{6} T^{6} + 1012 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 171 T - 151344 T^{2} + 121435957 T^{3} + 55436943534 T^{4} + 121435957 p^{3} T^{5} - 151344 p^{6} T^{6} - 171 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 1019 T + 1458508 T^{2} - 984898311 T^{3} + 770903521478 T^{4} - 984898311 p^{3} T^{5} + 1458508 p^{6} T^{6} - 1019 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 1836 T + 2484428 T^{2} + 2154571740 T^{3} + 1575960026870 T^{4} + 2154571740 p^{3} T^{5} + 2484428 p^{6} T^{6} + 1836 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1374 T + 1841980 T^{2} + 1551227238 T^{3} + 1329206295750 T^{4} + 1551227238 p^{3} T^{5} + 1841980 p^{6} T^{6} + 1374 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1248 T + 2235324 T^{2} - 1893641568 T^{3} + 1933448532790 T^{4} - 1893641568 p^{3} T^{5} + 2235324 p^{6} T^{6} - 1248 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1401 T + 2898526 T^{2} + 2521534159 T^{3} + 2998193496994 T^{4} + 2521534159 p^{3} T^{5} + 2898526 p^{6} T^{6} + 1401 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 2559 T + 5248530 T^{2} + 7235676241 T^{3} + 7900785018810 T^{4} + 7235676241 p^{3} T^{5} + 5248530 p^{6} T^{6} + 2559 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.14403616771503572068666187583, −6.92693644482862713669871464121, −6.41564424027374552298800437067, −6.31990944535217989253486084543, −6.29694694555893458261590785443, −5.71216644148081157837837020600, −5.51079988182363948492373920307, −5.37954988754626804859158186053, −5.30113099440602684013708855760, −5.05051263744825856867427957037, −4.69463853665594786460226571585, −4.40217565532555011205402417413, −4.26728259931913599816549965376, −3.99212615176713428861092334138, −3.64480502689088908395026436866, −3.57052650662523175794903289705, −3.55359816061404598140638027554, −2.79584288418432853341875950484, −2.57016942800066345120403038691, −2.53404367153357959049816068846, −2.51127029988340521525064447944, −1.66719200574624868612009237740, −1.62813375997084114162264097043, −1.33350334407364383200901178496, −0.971984813573814089143827776761, 0, 0, 0, 0, 0.971984813573814089143827776761, 1.33350334407364383200901178496, 1.62813375997084114162264097043, 1.66719200574624868612009237740, 2.51127029988340521525064447944, 2.53404367153357959049816068846, 2.57016942800066345120403038691, 2.79584288418432853341875950484, 3.55359816061404598140638027554, 3.57052650662523175794903289705, 3.64480502689088908395026436866, 3.99212615176713428861092334138, 4.26728259931913599816549965376, 4.40217565532555011205402417413, 4.69463853665594786460226571585, 5.05051263744825856867427957037, 5.30113099440602684013708855760, 5.37954988754626804859158186053, 5.51079988182363948492373920307, 5.71216644148081157837837020600, 6.29694694555893458261590785443, 6.31990944535217989253486084543, 6.41564424027374552298800437067, 6.92693644482862713669871464121, 7.14403616771503572068666187583

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