Properties

Label 8-48e4-1.1-c12e4-0-2
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $3.70457\times 10^{6}$
Root an. cond. $6.62357$
Motivic weight $12$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.19e4·5-s − 3.54e5·9-s − 5.81e6·13-s − 3.38e7·17-s − 1.76e8·25-s + 4.25e7·29-s + 9.40e9·37-s + 1.79e10·41-s + 7.78e9·45-s + 5.46e10·49-s + 6.05e10·53-s + 1.68e11·61-s + 1.27e11·65-s + 7.89e11·73-s + 9.41e10·81-s + 7.44e11·85-s − 6.38e11·89-s − 5.63e11·97-s − 4.63e12·101-s − 5.39e12·109-s − 9.04e12·113-s + 2.05e12·117-s + 4.12e12·121-s + 7.05e12·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.40·5-s − 2/3·9-s − 1.20·13-s − 1.40·17-s − 0.721·25-s + 0.0715·29-s + 3.66·37-s + 3.78·41-s + 0.936·45-s + 3.94·49-s + 2.73·53-s + 3.26·61-s + 1.69·65-s + 5.21·73-s + 1/3·81-s + 1.97·85-s − 1.28·89-s − 0.676·97-s − 4.37·101-s − 3.21·109-s − 4.34·113-s + 0.802·117-s + 1.31·121-s + 1.85·125-s − 0.100·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3.70457\times 10^{6}\)
Root analytic conductor: \(6.62357\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :6, 6, 6, 6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.793142960\)
\(L(\frac12)\) \(\approx\) \(2.793142960\)
\(L(7)\) 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 \( 1 \)
3$C_2$ \( ( 1 + p^{11} T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 + 2196 p T + 430214 p^{4} T^{2} + 2196 p^{13} T^{3} + p^{24} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 54652997284 T^{2} + 23057278827917426598 p^{2} T^{4} - 54652997284 p^{24} T^{6} + p^{48} T^{8} \)
11$D_4\times C_2$ \( 1 - 34088557060 p^{2} T^{2} + \)\(16\!\cdots\!02\)\( p^{4} T^{4} - 34088557060 p^{26} T^{6} + p^{48} T^{8} \)
13$D_{4}$ \( ( 1 + 2905036 T - 1825385214714 T^{2} + 2905036 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 16941132 T + 1004522979764774 T^{2} + 16941132 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 5818731155199844 T^{2} + \)\(15\!\cdots\!02\)\( T^{4} - 5818731155199844 p^{24} T^{6} + p^{48} T^{8} \)
23$D_4\times C_2$ \( 1 - 40658686053902212 T^{2} + \)\(84\!\cdots\!22\)\( T^{4} - 40658686053902212 p^{24} T^{6} + p^{48} T^{8} \)
29$D_{4}$ \( ( 1 - 21291948 T + 565340476276432262 T^{2} - 21291948 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 1971519050733147940 T^{2} + \)\(19\!\cdots\!42\)\( T^{4} - 1971519050733147940 p^{24} T^{6} + p^{48} T^{8} \)
37$D_{4}$ \( ( 1 - 4701432868 T + 17305228534658117862 T^{2} - 4701432868 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 8984441268 T + 63939568982503208294 T^{2} - 8984441268 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 99624310717809132004 T^{2} + \)\(56\!\cdots\!82\)\( T^{4} - 99624310717809132004 p^{24} T^{6} + p^{48} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(10\!\cdots\!04\)\( T^{2} - \)\(11\!\cdots\!18\)\( T^{4} - \)\(10\!\cdots\!04\)\( p^{24} T^{6} + p^{48} T^{8} \)
53$D_{4}$ \( ( 1 - 30273478380 T + \)\(10\!\cdots\!06\)\( T^{2} - 30273478380 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(25\!\cdots\!60\)\( T^{2} + \)\(33\!\cdots\!22\)\( T^{4} - \)\(25\!\cdots\!60\)\( p^{24} T^{6} + p^{48} T^{8} \)
61$D_{4}$ \( ( 1 - 84143600836 T + \)\(70\!\cdots\!62\)\( T^{2} - 84143600836 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(49\!\cdots\!04\)\( T^{2} + \)\(27\!\cdots\!22\)\( T^{4} - \)\(49\!\cdots\!04\)\( p^{24} T^{6} + p^{48} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(63\!\cdots\!84\)\( T^{2} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(63\!\cdots\!84\)\( p^{24} T^{6} + p^{48} T^{8} \)
73$D_{4}$ \( ( 1 - 394513314500 T + \)\(75\!\cdots\!46\)\( T^{2} - 394513314500 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!36\)\( T^{2} + \)\(12\!\cdots\!90\)\( T^{4} - \)\(14\!\cdots\!36\)\( p^{24} T^{6} + p^{48} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(37\!\cdots\!48\)\( T^{2} + \)\(56\!\cdots\!18\)\( T^{4} - \)\(37\!\cdots\!48\)\( p^{24} T^{6} + p^{48} T^{8} \)
89$D_{4}$ \( ( 1 + 319335230268 T + \)\(42\!\cdots\!98\)\( T^{2} + 319335230268 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 281771021500 T + \)\(26\!\cdots\!98\)\( T^{2} + 281771021500 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
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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

−9.278163756072993582234845795923, −8.449179953700750086459825559309, −8.249477848298314217456047707809, −7.984987763389579595616967227301, −7.937999728147007346316472955834, −7.35643010147094057975504996780, −7.03348738984483564798196120960, −6.89485868811338500152151196875, −6.46708496067575522072396924590, −5.82039906161990514585635480996, −5.63895316128618963457542009899, −5.44461989718939175091931234745, −4.94011357321322201048170286252, −4.15006887944541264230328198471, −4.08742554032965335842829706067, −4.02872488606677458696190180753, −3.87170926014751177166051518280, −2.65952765720757215377310393232, −2.54836108643428779291332230687, −2.43821137528930516635070170926, −2.29698603177724281578608353303, −1.20835996914797871039960536674, −0.74265847474827524918982195345, −0.66120921594699206878598908342, −0.32444443615239857451950420280, 0.32444443615239857451950420280, 0.66120921594699206878598908342, 0.74265847474827524918982195345, 1.20835996914797871039960536674, 2.29698603177724281578608353303, 2.43821137528930516635070170926, 2.54836108643428779291332230687, 2.65952765720757215377310393232, 3.87170926014751177166051518280, 4.02872488606677458696190180753, 4.08742554032965335842829706067, 4.15006887944541264230328198471, 4.94011357321322201048170286252, 5.44461989718939175091931234745, 5.63895316128618963457542009899, 5.82039906161990514585635480996, 6.46708496067575522072396924590, 6.89485868811338500152151196875, 7.03348738984483564798196120960, 7.35643010147094057975504996780, 7.937999728147007346316472955834, 7.984987763389579595616967227301, 8.249477848298314217456047707809, 8.449179953700750086459825559309, 9.278163756072993582234845795923

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