Properties

Label 8-48e4-1.1-c15e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $2.20080\times 10^{7}$
Root an. cond. $8.27604$
Motivic weight $15$
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  + 2.65e7·9-s − 6.65e8·13-s − 4.47e10·25-s − 1.25e12·37-s + 1.88e13·49-s + 8.62e11·61-s − 3.72e14·73-s + 5.01e14·81-s + 4.64e15·97-s − 1.21e16·109-s − 1.76e16·117-s − 6.92e15·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.20e16·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 1.85·9-s − 2.94·13-s − 1.46·25-s − 2.16·37-s + 3.96·49-s + 0.0351·61-s − 3.94·73-s + 2.43·81-s + 5.83·97-s − 6.37·109-s − 5.45·117-s − 1.65·121-s + 1.40·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+15/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.20080\times 10^{7}\)
Root analytic conductor: \(8.27604\)
Motivic weight: \(15\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :15/2, 15/2, 15/2, 15/2),\ 1)\)

Particular Values

\(L(8)\) \(\approx\) \(0.8477034184\)
\(L(\frac12)\) \(\approx\) \(0.8477034184\)
\(L(\frac{17}{2})\) 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 \( 1 \)
3$C_2^2$ \( 1 - 12158 p^{7} T^{2} + p^{30} T^{4} \)
good5$C_2^2$ \( ( 1 + 4472960494 p T^{2} + p^{30} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 192004967426 p^{2} T^{2} + p^{30} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 28628981943862 p^{2} T^{2} + p^{30} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 12798770 p T + p^{15} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 5621089645499631586 T^{2} + p^{30} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 12206075725639426102 T^{2} + p^{30} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + \)\(28\!\cdots\!74\)\( T^{2} + p^{30} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 9767706874797348058 p^{2} T^{2} + p^{30} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - \)\(17\!\cdots\!02\)\( T^{2} + p^{30} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 312995380930 T + p^{15} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - \)\(14\!\cdots\!22\)\( T^{2} + p^{30} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + \)\(20\!\cdots\!54\)\( T^{2} + p^{30} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + \)\(23\!\cdots\!46\)\( T^{2} + p^{30} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - \)\(47\!\cdots\!14\)\( T^{2} + p^{30} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + \)\(17\!\cdots\!98\)\( T^{2} + p^{30} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 215703610022 T + p^{15} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - \)\(50\!\cdots\!14\)\( T^{2} + p^{30} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + \)\(27\!\cdots\!02\)\( T^{2} + p^{30} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 93057038642630 T + p^{15} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + \)\(24\!\cdots\!02\)\( T^{2} + p^{30} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + \)\(91\!\cdots\!74\)\( T^{2} + p^{30} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + \)\(20\!\cdots\!82\)\( T^{2} + p^{30} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 1161599117677810 T + p^{15} T^{2} )^{4} \)
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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

−8.504656195613475843959435144497, −7.975746986133151140939998270464, −7.59722823622954473263711345578, −7.37153960677118351289800180523, −7.36659649191542395953743905561, −7.04092091709161678414436481286, −6.71187709157088657992119097746, −6.11349166107951734714574727515, −6.02184375445106118901029478405, −5.22957122773190915434889624995, −5.19395177264968080653779007854, −5.03437553809276506372673790637, −4.47256635661475717726134152949, −4.07240821620828522734339177165, −3.97358173930095709981306150940, −3.71079999219358485926395055974, −2.84183432448001881187597976157, −2.81683002235078083591104597365, −2.34300553080646390470324845899, −2.03382869901908628065287878941, −1.57126418192836727885298885118, −1.55816172379368858966013180051, −0.853261816116257571103935664940, −0.53341761960426422224667348169, −0.12544447055071092734529415151, 0.12544447055071092734529415151, 0.53341761960426422224667348169, 0.853261816116257571103935664940, 1.55816172379368858966013180051, 1.57126418192836727885298885118, 2.03382869901908628065287878941, 2.34300553080646390470324845899, 2.81683002235078083591104597365, 2.84183432448001881187597976157, 3.71079999219358485926395055974, 3.97358173930095709981306150940, 4.07240821620828522734339177165, 4.47256635661475717726134152949, 5.03437553809276506372673790637, 5.19395177264968080653779007854, 5.22957122773190915434889624995, 6.02184375445106118901029478405, 6.11349166107951734714574727515, 6.71187709157088657992119097746, 7.04092091709161678414436481286, 7.36659649191542395953743905561, 7.37153960677118351289800180523, 7.59722823622954473263711345578, 7.975746986133151140939998270464, 8.504656195613475843959435144497

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