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Properties

Label 8-75e4-1.1-c21e4-0-1
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $1.93032\times 10^{9}$
Root an. cond. $14.4778$
Motivic weight $21$
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  + 3.01e6·4-s − 6.97e9·9-s + 4.39e11·11-s − 8.35e11·16-s − 2.39e13·19-s + 3.59e15·29-s + 2.23e16·31-s − 2.10e16·36-s + 2.45e17·41-s + 1.32e18·44-s + 1.83e18·49-s + 7.04e18·59-s − 3.55e18·61-s − 1.91e19·64-s + 3.47e19·71-s − 7.21e19·76-s + 1.08e20·79-s + 3.64e19·81-s − 4.53e20·89-s − 3.06e21·99-s − 2.97e21·101-s + 3.25e20·109-s + 1.08e22·116-s + 9.62e22·121-s + 6.73e22·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.43·4-s − 2/3·9-s + 5.11·11-s − 0.189·16-s − 0.895·19-s + 1.58·29-s + 4.89·31-s − 0.958·36-s + 2.86·41-s + 7.34·44-s + 3.28·49-s + 1.79·59-s − 0.638·61-s − 2.08·64-s + 1.26·71-s − 1.28·76-s + 1.28·79-s + 1/3·81-s − 1.54·89-s − 3.40·99-s − 2.67·101-s + 0.131·109-s + 2.28·116-s + 13.0·121-s + 7.03·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+21/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.93032\times 10^{9}\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :21/2, 21/2, 21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(37.22768205\)
\(L(\frac12)\) \(\approx\) \(37.22768205\)
\(L(\frac{23}{2})\) 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)$
bad3$C_2$ \( ( 1 + p^{20} T^{2} )^{2} \)
5 \( 1 \)
good2$D_4\times C_2$ \( 1 - 753767 p^{2} T^{2} + 9693558897 p^{10} T^{4} - 753767 p^{44} T^{6} + p^{84} T^{8} \)
7$D_4\times C_2$ \( 1 - 37395451532817116 p^{2} T^{2} + \)\(59\!\cdots\!78\)\( p^{4} T^{4} - 37395451532817116 p^{44} T^{6} + p^{84} T^{8} \)
11$D_{4}$ \( ( 1 - 19988102088 p T + \)\(20\!\cdots\!54\)\( p^{2} T^{2} - 19988102088 p^{22} T^{3} + p^{42} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + \)\(17\!\cdots\!68\)\( T^{2} + \)\(76\!\cdots\!82\)\( p^{2} T^{4} + \)\(17\!\cdots\!68\)\( p^{42} T^{6} + p^{84} T^{8} \)
17$D_4\times C_2$ \( 1 - \)\(68\!\cdots\!80\)\( p^{2} T^{2} + \)\(22\!\cdots\!18\)\( p^{4} T^{4} - \)\(68\!\cdots\!80\)\( p^{44} T^{6} + p^{84} T^{8} \)
19$D_{4}$ \( ( 1 + 629504474296 p T + \)\(48\!\cdots\!38\)\( p^{2} T^{2} + 629504474296 p^{22} T^{3} + p^{42} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!76\)\( T^{2} + \)\(63\!\cdots\!38\)\( T^{4} - \)\(11\!\cdots\!76\)\( p^{42} T^{6} + p^{84} T^{8} \)
29$D_{4}$ \( ( 1 - 1798520043674052 T + \)\(75\!\cdots\!58\)\( T^{2} - 1798520043674052 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 11169107526944992 T + \)\(65\!\cdots\!62\)\( T^{2} - 11169107526944992 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - \)\(21\!\cdots\!64\)\( T^{2} + \)\(25\!\cdots\!98\)\( T^{4} - \)\(21\!\cdots\!64\)\( p^{42} T^{6} + p^{84} T^{8} \)
41$D_{4}$ \( ( 1 - 122972020616468052 T + \)\(11\!\cdots\!02\)\( T^{2} - 122972020616468052 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(37\!\cdots\!60\)\( T^{2} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(37\!\cdots\!60\)\( p^{42} T^{6} + p^{84} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(16\!\cdots\!20\)\( T^{2} + \)\(40\!\cdots\!18\)\( T^{4} - \)\(16\!\cdots\!20\)\( p^{42} T^{6} + p^{84} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(60\!\cdots\!96\)\( T^{2} + \)\(14\!\cdots\!58\)\( T^{4} - \)\(60\!\cdots\!96\)\( p^{42} T^{6} + p^{84} T^{8} \)
59$D_{4}$ \( ( 1 - 3523823330903857224 T + \)\(22\!\cdots\!98\)\( T^{2} - 3523823330903857224 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 1779023128451013860 T + \)\(54\!\cdots\!38\)\( T^{2} + 1779023128451013860 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(69\!\cdots\!60\)\( T^{2} + \)\(21\!\cdots\!78\)\( T^{4} - \)\(69\!\cdots\!60\)\( p^{42} T^{6} + p^{84} T^{8} \)
71$D_{4}$ \( ( 1 - 17379227131150420944 T + \)\(15\!\cdots\!26\)\( T^{2} - 17379227131150420944 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(61\!\cdots\!32\)\( p T^{2} - \)\(10\!\cdots\!02\)\( T^{4} - \)\(61\!\cdots\!32\)\( p^{43} T^{6} + p^{84} T^{8} \)
79$D_{4}$ \( ( 1 - 54055785594190591040 T + \)\(14\!\cdots\!58\)\( T^{2} - 54055785594190591040 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(67\!\cdots\!12\)\( T^{2} + \)\(19\!\cdots\!58\)\( T^{4} - \)\(67\!\cdots\!12\)\( p^{42} T^{6} + p^{84} T^{8} \)
89$D_{4}$ \( ( 1 + \)\(22\!\cdots\!24\)\( T + \)\(17\!\cdots\!98\)\( T^{2} + \)\(22\!\cdots\!24\)\( p^{21} T^{3} + p^{42} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!40\)\( T^{2} + \)\(11\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!40\)\( p^{42} T^{6} + p^{84} T^{8} \)
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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

−6.79827035080618157836235437919, −6.69224736344413004036323096692, −6.63739855544251972700228819658, −6.29640882822115764512252909217, −6.11961512970861916435295277127, −5.86240543607005187397095327084, −5.61023175763803207998288108679, −4.88870707684333374631171521383, −4.62767890489364007231218704559, −4.35021964433810738816001787369, −4.20822077369177752457945192586, −4.00481222297773563482225710949, −3.67532065460308818396924127690, −3.51816363174516011300374701422, −2.84952277548593587170661003060, −2.56336081704307637713463716147, −2.47537815326291899209478447555, −2.42348668898736925595271242704, −2.01538143135935146631738753796, −1.25251227757390832482223488218, −1.23976188385864166976066452703, −1.17307038601411434892283850991, −1.06126423226941952388887175361, −0.56104603174521494499067560796, −0.40058697550043657138456475694, 0.40058697550043657138456475694, 0.56104603174521494499067560796, 1.06126423226941952388887175361, 1.17307038601411434892283850991, 1.23976188385864166976066452703, 1.25251227757390832482223488218, 2.01538143135935146631738753796, 2.42348668898736925595271242704, 2.47537815326291899209478447555, 2.56336081704307637713463716147, 2.84952277548593587170661003060, 3.51816363174516011300374701422, 3.67532065460308818396924127690, 4.00481222297773563482225710949, 4.20822077369177752457945192586, 4.35021964433810738816001787369, 4.62767890489364007231218704559, 4.88870707684333374631171521383, 5.61023175763803207998288108679, 5.86240543607005187397095327084, 6.11961512970861916435295277127, 6.29640882822115764512252909217, 6.63739855544251972700228819658, 6.69224736344413004036323096692, 6.79827035080618157836235437919

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