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Properties

Label 8-6762e4-1.1-c1e4-0-11
Degree $8$
Conductor $2.091\times 10^{15}$
Sign $1$
Analytic cond. $8.49980\times 10^{6}$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 4·3-s + 10·4-s − 8·5-s − 16·6-s + 20·8-s + 10·9-s − 32·10-s + 4·11-s − 40·12-s − 12·13-s + 32·15-s + 35·16-s + 4·17-s + 40·18-s − 4·19-s − 80·20-s + 16·22-s + 4·23-s − 80·24-s + 24·25-s − 48·26-s − 20·27-s + 8·29-s + 128·30-s + 4·31-s + 56·32-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s − 3.57·5-s − 6.53·6-s + 7.07·8-s + 10/3·9-s − 10.1·10-s + 1.20·11-s − 11.5·12-s − 3.32·13-s + 8.26·15-s + 35/4·16-s + 0.970·17-s + 9.42·18-s − 0.917·19-s − 17.8·20-s + 3.41·22-s + 0.834·23-s − 16.3·24-s + 24/5·25-s − 9.41·26-s − 3.84·27-s + 1.48·29-s + 23.3·30-s + 0.718·31-s + 9.89·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(8.49980\times 10^{6}\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
3$C_1$ \( ( 1 + T )^{4} \)
7 \( 1 \)
23$C_1$ \( ( 1 - T )^{4} \)
good5$D_{4}$ \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 24 T^{2} - 84 T^{3} + 302 T^{4} - 84 p T^{5} + 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 88 T^{2} + 452 T^{3} + 1838 T^{4} + 452 p T^{5} + 88 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 56 T^{2} - 156 T^{3} + 1310 T^{4} - 156 p T^{5} + 56 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 20 T^{2} - 52 T^{3} - 254 T^{4} - 52 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 12 T^{2} + 40 T^{3} - 202 T^{4} + 40 p T^{5} + 12 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 68 T^{2} - 412 T^{3} + 2322 T^{4} - 412 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 136 T^{2} - 396 T^{3} + 7310 T^{4} - 396 p T^{5} + 136 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 80 T^{2} - 8 p T^{3} + 2722 T^{4} - 8 p^{2} T^{5} + 80 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 48 T^{2} - 404 T^{3} + 3518 T^{4} - 404 p T^{5} + 48 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 132 T^{2} + 980 T^{3} + 8402 T^{4} + 980 p T^{5} + 132 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 120 T^{2} - 364 T^{3} + 8750 T^{4} - 364 p T^{5} + 120 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 204 T^{2} + 1808 T^{3} + 16086 T^{4} + 1808 p T^{5} + 204 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 232 T^{2} + 2576 T^{3} + 20866 T^{4} + 2576 p T^{5} + 232 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 304 T^{2} - 3460 T^{3} + 31774 T^{4} - 3460 p T^{5} + 304 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 444 T^{2} + 5176 T^{3} + 51542 T^{4} + 5176 p T^{5} + 444 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 224 T^{2} + 1288 T^{3} + 21730 T^{4} + 1288 p T^{5} + 224 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 32 T + 612 T^{2} + 7840 T^{3} + 1002 p T^{4} + 7840 p T^{5} + 612 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 308 T^{2} - 1004 T^{3} + 37378 T^{4} - 1004 p T^{5} + 308 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 168 T^{2} + 684 T^{3} + 46 p T^{4} + 684 p T^{5} + 168 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 184 T^{2} + 636 T^{3} + 24830 T^{4} + 636 p T^{5} + 184 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.99799656422372051308610845716, −5.56832530938695419572344297698, −5.52402225011793311158951394242, −5.45122783858278358984157429859, −5.06587564701701859870859289862, −4.89940380347789854185914907135, −4.73410228086790106923141370982, −4.65062309420377419945812884954, −4.53803835243252435625748274658, −4.24925659638848762270074605696, −4.18308163819311210555428201010, −4.15827723038528593900367427539, −3.94865612432082169104247467268, −3.65751830233570174152884135918, −3.37846511806629529318601541079, −3.24416573618782249956792631155, −3.10955300751289579886479977648, −2.63248154481851124218805999272, −2.55227879375180795696251839298, −2.41701903562882615292488562165, −2.29860442363559206205388940083, −1.41845958538450168867153725212, −1.31404839588862579550890093529, −1.21437916918972185492028579897, −1.14305165865844566826536643843, 0, 0, 0, 0, 1.14305165865844566826536643843, 1.21437916918972185492028579897, 1.31404839588862579550890093529, 1.41845958538450168867153725212, 2.29860442363559206205388940083, 2.41701903562882615292488562165, 2.55227879375180795696251839298, 2.63248154481851124218805999272, 3.10955300751289579886479977648, 3.24416573618782249956792631155, 3.37846511806629529318601541079, 3.65751830233570174152884135918, 3.94865612432082169104247467268, 4.15827723038528593900367427539, 4.18308163819311210555428201010, 4.24925659638848762270074605696, 4.53803835243252435625748274658, 4.65062309420377419945812884954, 4.73410228086790106923141370982, 4.89940380347789854185914907135, 5.06587564701701859870859289862, 5.45122783858278358984157429859, 5.52402225011793311158951394242, 5.56832530938695419572344297698, 5.99799656422372051308610845716

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