Properties

Label 8-2e40-1.1-c1e4-0-10
Degree $8$
Conductor $1.100\times 10^{12}$
Sign $1$
Analytic cond. $4470.00$
Root an. cond. $2.85948$
Motivic weight $1$
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  + 8·5-s − 4·9-s + 8·13-s + 24·25-s + 24·29-s + 8·37-s − 16·41-s − 32·45-s − 12·49-s + 24·53-s + 24·61-s + 64·65-s + 8·73-s + 2·81-s + 8·89-s + 32·97-s + 40·101-s − 24·109-s − 8·113-s − 32·117-s − 4·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + 192·145-s + ⋯
L(s)  = 1  + 3.57·5-s − 4/3·9-s + 2.21·13-s + 24/5·25-s + 4.45·29-s + 1.31·37-s − 2.49·41-s − 4.77·45-s − 1.71·49-s + 3.29·53-s + 3.07·61-s + 7.93·65-s + 0.936·73-s + 2/9·81-s + 0.847·89-s + 3.24·97-s + 3.98·101-s − 2.29·109-s − 0.752·113-s − 2.95·117-s − 0.363·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 15.9·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40}\)
Sign: $1$
Analytic conductor: \(4470.00\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1024} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{40} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.22171440\)
\(L(\frac12)\) \(\approx\) \(10.22171440\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2:C_4$ \( 1 + 4 T^{2} + 14 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
5$D_{4}$ \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
7$C_2^2:C_4$ \( 1 + 12 T^{2} + 102 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2:C_4$ \( 1 + 4 T^{2} + 238 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2:C_4$ \( 1 + 36 T^{2} + 654 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2:C_4$ \( 1 + 12 T^{2} + 1062 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2:C_4$ \( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
43$C_2^2:C_4$ \( 1 + 164 T^{2} + 10414 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2:C_4$ \( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 12 T + 140 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2:C_4$ \( 1 + 228 T^{2} + 19950 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 12 T + 108 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2:C_4$ \( 1 + 164 T^{2} + 13390 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2:C_4$ \( 1 + 76 T^{2} + 9958 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2:C_4$ \( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 + 132 T^{2} + 10446 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.71760415623890164988738747847, −6.68372788424544699424761403689, −6.56165961618439836198526720966, −6.38157768700889576918052761335, −6.33969367723584567305766741506, −5.82626464540961838981061684282, −5.82120843789372009944298462339, −5.73579771362863886383036241019, −5.48868131257463684832976022267, −5.00277835079804044307026768914, −4.99352124326692809550077641306, −4.80382925134991968340577962083, −4.47773709437678088260033584226, −3.89480476744320597052986510397, −3.78169473542701537862071289041, −3.39863272147648784594467661309, −3.30048009715135976557459575386, −2.79937862571030686779798035062, −2.42991404686427314299510988882, −2.37617883816608787264309684634, −2.22630838807908370909208512995, −1.73155908422959962028865206419, −1.39136315239127214852981826771, −0.980969940577510576536911232947, −0.77928065834658055078697526035, 0.77928065834658055078697526035, 0.980969940577510576536911232947, 1.39136315239127214852981826771, 1.73155908422959962028865206419, 2.22630838807908370909208512995, 2.37617883816608787264309684634, 2.42991404686427314299510988882, 2.79937862571030686779798035062, 3.30048009715135976557459575386, 3.39863272147648784594467661309, 3.78169473542701537862071289041, 3.89480476744320597052986510397, 4.47773709437678088260033584226, 4.80382925134991968340577962083, 4.99352124326692809550077641306, 5.00277835079804044307026768914, 5.48868131257463684832976022267, 5.73579771362863886383036241019, 5.82120843789372009944298462339, 5.82626464540961838981061684282, 6.33969367723584567305766741506, 6.38157768700889576918052761335, 6.56165961618439836198526720966, 6.68372788424544699424761403689, 6.71760415623890164988738747847

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