Properties

Label 8-50e4-1.1-c21e4-0-0
Degree $8$
Conductor $6250000$
Sign $1$
Analytic cond. $3.81299\times 10^{8}$
Root an. cond. $11.8211$
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  − 2.09e6·4-s + 9.05e9·9-s − 8.26e10·11-s + 3.29e12·16-s − 1.43e13·19-s + 4.90e15·29-s − 8.62e15·31-s − 1.89e16·36-s − 3.25e16·41-s + 1.73e17·44-s + 8.65e17·49-s − 6.22e18·59-s − 1.58e18·61-s − 4.61e18·64-s − 1.64e20·71-s + 3.00e19·76-s − 5.66e19·79-s + 4.13e19·81-s − 9.57e20·89-s − 7.48e20·99-s − 2.32e20·101-s + 7.60e21·109-s − 1.02e22·116-s − 2.80e21·121-s + 1.80e22·124-s + ⋯
L(s)  = 1  − 4-s + 0.865·9-s − 0.960·11-s + 3/4·16-s − 0.537·19-s + 2.16·29-s − 1.88·31-s − 0.865·36-s − 0.379·41-s + 0.960·44-s + 1.54·49-s − 1.58·59-s − 0.284·61-s − 1/2·64-s − 6.00·71-s + 0.537·76-s − 0.672·79-s + 0.378·81-s − 3.25·89-s − 0.831·99-s − 0.209·101-s + 3.07·109-s − 2.16·116-s − 0.378·121-s + 1.88·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s+21/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(6250000\)    =    \(2^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3.81299\times 10^{8}\)
Root analytic conductor: \(11.8211\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 6250000,\ (\ :21/2, 21/2, 21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(0.0006859159437\)
\(L(\frac12)\) \(\approx\) \(0.0006859159437\)
\(L(\frac{23}{2})\) 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)$
bad2$C_2$ \( ( 1 + p^{20} T^{2} )^{2} \)
5 \( 1 \)
good3$D_4\times C_2$ \( 1 - 9053968580 T^{2} + 55700588724588742 p^{6} T^{4} - 9053968580 p^{42} T^{6} + p^{84} T^{8} \)
7$D_4\times C_2$ \( 1 - 17666443091494100 p^{2} T^{2} + \)\(31\!\cdots\!98\)\( p^{4} T^{4} - 17666443091494100 p^{44} T^{6} + p^{84} T^{8} \)
11$D_{4}$ \( ( 1 + 41326831776 T + \)\(36\!\cdots\!06\)\( p T^{2} + 41326831776 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + \)\(46\!\cdots\!60\)\( p^{2} T^{2} + \)\(96\!\cdots\!58\)\( p^{4} T^{4} + \)\(46\!\cdots\!60\)\( p^{44} T^{6} + p^{84} T^{8} \)
17$D_4\times C_2$ \( 1 - \)\(65\!\cdots\!20\)\( T^{2} + \)\(33\!\cdots\!02\)\( p^{2} T^{4} - \)\(65\!\cdots\!20\)\( p^{42} T^{6} + p^{84} T^{8} \)
19$D_{4}$ \( ( 1 + 377673402760 p T + \)\(36\!\cdots\!58\)\( p^{2} T^{2} + 377673402760 p^{22} T^{3} + p^{42} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - \)\(47\!\cdots\!40\)\( T^{2} + \)\(72\!\cdots\!58\)\( T^{4} - \)\(47\!\cdots\!40\)\( p^{42} T^{6} + p^{84} T^{8} \)
29$D_{4}$ \( ( 1 - 2454584202185940 T + \)\(46\!\cdots\!58\)\( T^{2} - 2454584202185940 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 4310677151243336 T + \)\(16\!\cdots\!86\)\( T^{2} + 4310677151243336 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!60\)\( T^{2} + \)\(11\!\cdots\!38\)\( T^{4} - \)\(14\!\cdots\!60\)\( p^{42} T^{6} + p^{84} T^{8} \)
41$D_{4}$ \( ( 1 + 16298035212712116 T + \)\(71\!\cdots\!46\)\( T^{2} + 16298035212712116 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(67\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!98\)\( T^{4} - \)\(67\!\cdots\!00\)\( p^{42} T^{6} + p^{84} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(35\!\cdots\!80\)\( T^{2} + \)\(62\!\cdots\!18\)\( T^{4} - \)\(35\!\cdots\!80\)\( p^{42} T^{6} + p^{84} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(37\!\cdots\!80\)\( T^{2} + \)\(84\!\cdots\!18\)\( T^{4} - \)\(37\!\cdots\!80\)\( p^{42} T^{6} + p^{84} T^{8} \)
59$D_{4}$ \( ( 1 + 3110197357974672120 T + \)\(32\!\cdots\!18\)\( T^{2} + 3110197357974672120 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 791142209760451676 T + \)\(37\!\cdots\!66\)\( T^{2} + 791142209760451676 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(65\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!78\)\( T^{4} - \)\(65\!\cdots\!20\)\( p^{42} T^{6} + p^{84} T^{8} \)
71$D_{4}$ \( ( 1 + 82388741313238482456 T + \)\(30\!\cdots\!26\)\( T^{2} + 82388741313238482456 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(53\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!58\)\( T^{4} - \)\(53\!\cdots\!40\)\( p^{42} T^{6} + p^{84} T^{8} \)
79$D_{4}$ \( ( 1 + 28314166125451369360 T + \)\(14\!\cdots\!58\)\( T^{2} + 28314166125451369360 p^{21} T^{3} + p^{42} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(76\!\cdots\!20\)\( T^{2} + \)\(22\!\cdots\!78\)\( T^{4} - \)\(76\!\cdots\!20\)\( p^{42} T^{6} + p^{84} T^{8} \)
89$D_{4}$ \( ( 1 + \)\(47\!\cdots\!80\)\( T + \)\(23\!\cdots\!78\)\( T^{2} + \)\(47\!\cdots\!80\)\( p^{21} T^{3} + p^{42} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!80\)\( T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!80\)\( p^{42} T^{6} + p^{84} T^{8} \)
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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

−7.54417252636714875647257965548, −7.31040638030936086077765291846, −7.23730730452418556431994255181, −6.56350959882920939534522970272, −6.43046666366326586317346366585, −5.87460520655683148254581752535, −5.73129483122729463667874221610, −5.61854849508492548553958271154, −4.99743225430063739331878990306, −4.74810983611518853019230784575, −4.54173793836631963184298455806, −4.43335434554314663760205284043, −3.95005329015847195158528765237, −3.72913823394362250626411401724, −3.34037971709274239522858428802, −2.92716915608561409333530049197, −2.78200163797293416087360571467, −2.42762114164923525205461769265, −2.07285163365818190915722848946, −1.61350479329723304815596916179, −1.30605383770036417843164441849, −1.12818701742824319067264261484, −0.976518162360805630247710172154, −0.14147877934534836486653220296, −0.008876411050267002943601506958, 0.008876411050267002943601506958, 0.14147877934534836486653220296, 0.976518162360805630247710172154, 1.12818701742824319067264261484, 1.30605383770036417843164441849, 1.61350479329723304815596916179, 2.07285163365818190915722848946, 2.42762114164923525205461769265, 2.78200163797293416087360571467, 2.92716915608561409333530049197, 3.34037971709274239522858428802, 3.72913823394362250626411401724, 3.95005329015847195158528765237, 4.43335434554314663760205284043, 4.54173793836631963184298455806, 4.74810983611518853019230784575, 4.99743225430063739331878990306, 5.61854849508492548553958271154, 5.73129483122729463667874221610, 5.87460520655683148254581752535, 6.43046666366326586317346366585, 6.56350959882920939534522970272, 7.23730730452418556431994255181, 7.31040638030936086077765291846, 7.54417252636714875647257965548

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