Properties

Label 8-2450e4-1.1-c1e4-0-0
Degree $8$
Conductor $3.603\times 10^{13}$
Sign $1$
Analytic cond. $146478.$
Root an. cond. $4.42304$
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  − 2·4-s + 8·9-s − 8·11-s + 3·16-s − 8·29-s − 16·36-s + 16·44-s − 4·64-s − 48·71-s + 16·79-s + 30·81-s − 64·99-s + 8·109-s + 16·116-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + ⋯
L(s)  = 1  − 4-s + 8/3·9-s − 2.41·11-s + 3/4·16-s − 1.48·29-s − 8/3·36-s + 2.41·44-s − 1/2·64-s − 5.69·71-s + 1.80·79-s + 10/3·81-s − 6.43·99-s + 0.766·109-s + 1.48·116-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(146478.\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5757554965\)
\(L(\frac12)\) \(\approx\) \(0.5757554965\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 144 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 96 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.25870595984772237712638429253, −6.10892993486972300448245676158, −6.10134708326366342986147386797, −5.61250838183266699608775584898, −5.39703501961417961774565221021, −5.22530009625841453639811876980, −5.13863499542476217145821917821, −4.81211703437517507733717847299, −4.58343787011781304545844076685, −4.58339779528212415410226986761, −4.37959716935488041952990505160, −3.88394352365241424384130695459, −3.77577075677783212847261880262, −3.70793225680344010697874165662, −3.54127434048832595028844361001, −2.93509116022770452964376302475, −2.75728794516669590052866321641, −2.59313551648394066405374158158, −2.42948268927100155239251657829, −1.83587082949540922336602752170, −1.66740413169707232129257630282, −1.39622223497822644934781913857, −1.24293490218134664069690712325, −0.59385065510202724474665624398, −0.15958919310640893485893370781, 0.15958919310640893485893370781, 0.59385065510202724474665624398, 1.24293490218134664069690712325, 1.39622223497822644934781913857, 1.66740413169707232129257630282, 1.83587082949540922336602752170, 2.42948268927100155239251657829, 2.59313551648394066405374158158, 2.75728794516669590052866321641, 2.93509116022770452964376302475, 3.54127434048832595028844361001, 3.70793225680344010697874165662, 3.77577075677783212847261880262, 3.88394352365241424384130695459, 4.37959716935488041952990505160, 4.58339779528212415410226986761, 4.58343787011781304545844076685, 4.81211703437517507733717847299, 5.13863499542476217145821917821, 5.22530009625841453639811876980, 5.39703501961417961774565221021, 5.61250838183266699608775584898, 6.10134708326366342986147386797, 6.10892993486972300448245676158, 6.25870595984772237712638429253

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