Properties

Label 8-882e4-1.1-c3e4-0-10
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $7.33396\times 10^{6}$
Root an. cond. $7.21385$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·4-s + 12·5-s − 16·8-s + 48·10-s + 4·11-s + 96·13-s − 64·16-s + 132·17-s − 120·19-s + 48·20-s + 16·22-s − 76·23-s + 284·25-s + 384·26-s + 224·29-s − 432·31-s − 64·32-s + 528·34-s + 280·37-s − 480·38-s − 192·40-s − 72·41-s − 256·43-s + 16·44-s − 304·46-s − 264·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 1.07·5-s − 0.707·8-s + 1.51·10-s + 0.109·11-s + 2.04·13-s − 16-s + 1.88·17-s − 1.44·19-s + 0.536·20-s + 0.155·22-s − 0.689·23-s + 2.27·25-s + 2.89·26-s + 1.43·29-s − 2.50·31-s − 0.353·32-s + 2.66·34-s + 1.24·37-s − 2.04·38-s − 0.758·40-s − 0.274·41-s − 0.907·43-s + 0.0548·44-s − 0.974·46-s − 0.819·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(7.33396\times 10^{6}\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(17.92404289\)
\(L(\frac12)\) \(\approx\) \(17.92404289\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 12 T - 28 p T^{2} - 408 T^{3} + 47031 T^{4} - 408 p^{3} T^{5} - 28 p^{7} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 4 T - 2578 T^{2} + 272 T^{3} + 4935979 T^{4} + 272 p^{3} T^{5} - 2578 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 48 T + 4520 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 132 T + 3484 T^{2} - 31944 p T^{3} + 292671 p^{2} T^{4} - 31944 p^{4} T^{5} + 3484 p^{6} T^{6} - 132 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 + 120 T + 1314 T^{2} - 75840 T^{3} + 25427915 T^{4} - 75840 p^{3} T^{5} + 1314 p^{6} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 + 76 T + 806 T^{2} - 1471664 T^{3} - 193611581 T^{4} - 1471664 p^{3} T^{5} + 806 p^{6} T^{6} + 76 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 - 112 T + 6914 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 432 T + 86218 T^{2} + 17635968 T^{3} + 3634145571 T^{4} + 17635968 p^{3} T^{5} + 86218 p^{6} T^{6} + 432 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 - 280 T + 22294 T^{2} + 12656000 T^{3} - 3389038373 T^{4} + 12656000 p^{3} T^{5} + 22294 p^{6} T^{6} - 280 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 36 T + 105908 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 128 T - 2778 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 264 T + 73114 T^{2} - 55720896 T^{3} - 18003580413 T^{4} - 55720896 p^{3} T^{5} + 73114 p^{6} T^{6} + 264 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 268 T - 170158 T^{2} + 14946896 T^{3} + 25697985547 T^{4} + 14946896 p^{3} T^{5} - 170158 p^{6} T^{6} - 268 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 336 T - 148478 T^{2} + 50193024 T^{3} + 2949366651 T^{4} + 50193024 p^{3} T^{5} - 148478 p^{6} T^{6} - 336 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 - 504 T - 14232 T^{2} + 93599856 T^{3} - 37220190553 T^{4} + 93599856 p^{3} T^{5} - 14232 p^{6} T^{6} - 504 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 384 T - 449462 T^{2} + 1769472 T^{3} + 221503407627 T^{4} + 1769472 p^{3} T^{5} - 449462 p^{6} T^{6} - 384 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 396 T + 521098 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 312 T + 3024 T^{2} + 213318768 T^{3} - 180306434737 T^{4} + 213318768 p^{3} T^{5} + 3024 p^{6} T^{6} - 312 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 - 848 T - 266750 T^{2} + 189952 T^{3} + 374274336739 T^{4} + 189952 p^{3} T^{5} - 266750 p^{6} T^{6} - 848 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 648 T + 1235750 T^{2} - 648 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 612 T - 1067780 T^{2} + 19820232 T^{3} + 1319275320879 T^{4} + 19820232 p^{3} T^{5} - 1067780 p^{6} T^{6} + 612 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 + 2184 T + 2982432 T^{2} + 2184 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−6.73361919148877356374227254377, −6.61892309873185445637982775011, −6.46949927214792239637585303120, −5.92684813783022966438984052132, −5.90668238405738728386918513130, −5.60154878452368981526220408082, −5.60004875572821065928934170306, −5.27449437201059194671885137545, −5.03295910020243852627312130830, −4.83390883946067356174364271003, −4.42067149340896726180764295017, −4.26045665583185082991370521831, −4.00214136728707687043970936329, −3.62174416181199247041007790572, −3.61427764118577519899886763179, −3.23177566150900069261710112847, −3.09041863033130340418872448228, −2.73138084125409056618180442270, −2.24409869367184915296051801650, −2.10522189486529417934661398755, −1.78800820376696563423529087367, −1.29437710147890435804708438873, −1.08230766597919244876640263595, −0.73444805336009011881767031159, −0.36588212544097549556219402180, 0.36588212544097549556219402180, 0.73444805336009011881767031159, 1.08230766597919244876640263595, 1.29437710147890435804708438873, 1.78800820376696563423529087367, 2.10522189486529417934661398755, 2.24409869367184915296051801650, 2.73138084125409056618180442270, 3.09041863033130340418872448228, 3.23177566150900069261710112847, 3.61427764118577519899886763179, 3.62174416181199247041007790572, 4.00214136728707687043970936329, 4.26045665583185082991370521831, 4.42067149340896726180764295017, 4.83390883946067356174364271003, 5.03295910020243852627312130830, 5.27449437201059194671885137545, 5.60004875572821065928934170306, 5.60154878452368981526220408082, 5.90668238405738728386918513130, 5.92684813783022966438984052132, 6.46949927214792239637585303120, 6.61892309873185445637982775011, 6.73361919148877356374227254377

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