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Properties

Label 8-9800e4-1.1-c1e4-0-0
Degree $8$
Conductor $9.224\times 10^{15}$
Sign $1$
Analytic cond. $3.74983\times 10^{7}$
Root an. cond. $8.84609$
Motivic weight $1$
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  − 2·3-s − 9-s + 2·11-s − 10·13-s − 6·17-s + 4·23-s + 2·27-s − 2·29-s − 12·31-s − 4·33-s + 20·39-s + 12·41-s + 8·43-s + 2·47-s + 12·51-s + 4·53-s + 8·59-s + 20·61-s + 8·67-s − 8·69-s + 4·71-s − 16·73-s + 22·79-s − 5·81-s − 36·83-s + 4·87-s + 40·89-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/3·9-s + 0.603·11-s − 2.77·13-s − 1.45·17-s + 0.834·23-s + 0.384·27-s − 0.371·29-s − 2.15·31-s − 0.696·33-s + 3.20·39-s + 1.87·41-s + 1.21·43-s + 0.291·47-s + 1.68·51-s + 0.549·53-s + 1.04·59-s + 2.56·61-s + 0.977·67-s − 0.963·69-s + 0.474·71-s − 1.87·73-s + 2.47·79-s − 5/9·81-s − 3.95·83-s + 0.428·87-s + 4.23·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.74983\times 10^{7}\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.400009633\)
\(L(\frac12)\) \(\approx\) \(1.400009633\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 26 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 3 p T^{2} - 46 T^{3} + 480 T^{4} - 46 p T^{5} + 3 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 71 T^{2} + 2 p^{2} T^{3} + 1384 T^{4} + 2 p^{3} T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + p T^{2} + 62 T^{3} + 434 T^{4} + 62 p T^{5} + p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 50 T^{2} + 24 T^{3} + 1186 T^{4} + 24 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 54 T^{2} - 324 T^{3} + 1442 T^{4} - 324 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 73 T^{2} + 154 T^{3} + 2740 T^{4} + 154 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 134 T^{2} + 828 T^{3} + 5602 T^{4} + 828 p T^{5} + 134 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 106 T^{2} + 40 T^{3} + 5114 T^{4} + 40 p T^{5} + 106 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 68 T^{2} + 52 T^{3} - 2014 T^{4} + 52 p T^{5} + 68 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 124 T^{2} - 840 T^{3} + 7718 T^{4} - 840 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 147 T^{2} - 378 T^{3} + 9344 T^{4} - 378 p T^{5} + 147 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + 8866 T^{4} - 4 p^{2} T^{5} + 142 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 288 T^{2} - 2572 T^{3} + 22350 T^{4} - 2572 p T^{5} + 288 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 154 T^{2} - 1176 T^{3} + 13994 T^{4} - 1176 p T^{5} + 154 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 32 T^{2} - 12 p T^{3} + 5438 T^{4} - 12 p^{2} T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 122 T^{2} - 240 T^{3} - 7038 T^{4} - 240 p T^{5} + 122 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 389 T^{2} - 4726 T^{3} + 48924 T^{4} - 4726 p T^{5} + 389 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 36 T + 750 T^{2} + 10564 T^{3} + 110978 T^{4} + 10564 p T^{5} + 750 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 40 T + 914 T^{2} - 13800 T^{3} + 152258 T^{4} - 13800 p T^{5} + 914 p^{2} T^{6} - 40 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 26 T + 545 T^{2} + 7602 T^{3} + 85890 T^{4} + 7602 p T^{5} + 545 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.44762041610938237810145920547, −5.13058754545201377783456729118, −5.02182728441768195623309772759, −4.99034649023686700551268172482, −4.76012531650647771736467513749, −4.41580967130799182140356191892, −4.23329856000612355624231901898, −4.22907651994938594222289302405, −3.99243840313051562083134055935, −3.83969222718931369317276399642, −3.48540592646756121221227346997, −3.34012209010765063006355069390, −3.17970564826847972181222906333, −2.77131975209990943993911132757, −2.61603618831014833338844107667, −2.45448963101901344211796892440, −2.29650170205009635545384436804, −2.17851635276973338364078060416, −1.84682908632839335613690123278, −1.65758316148225438297877059767, −1.42498030292193062747728960259, −0.71238887731321545964268806259, −0.65912870102371375118956343843, −0.59991464541068280961916287352, −0.22580827354147856782682099447, 0.22580827354147856782682099447, 0.59991464541068280961916287352, 0.65912870102371375118956343843, 0.71238887731321545964268806259, 1.42498030292193062747728960259, 1.65758316148225438297877059767, 1.84682908632839335613690123278, 2.17851635276973338364078060416, 2.29650170205009635545384436804, 2.45448963101901344211796892440, 2.61603618831014833338844107667, 2.77131975209990943993911132757, 3.17970564826847972181222906333, 3.34012209010765063006355069390, 3.48540592646756121221227346997, 3.83969222718931369317276399642, 3.99243840313051562083134055935, 4.22907651994938594222289302405, 4.23329856000612355624231901898, 4.41580967130799182140356191892, 4.76012531650647771736467513749, 4.99034649023686700551268172482, 5.02182728441768195623309772759, 5.13058754545201377783456729118, 5.44762041610938237810145920547

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