Properties

Label 8-1200e4-1.1-c1e4-0-21
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $8430.11$
Root an. cond. $3.09548$
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  + 8·13-s − 8·37-s + 16·49-s + 32·61-s + 56·73-s − 9·81-s − 40·97-s + 16·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2.21·13-s − 1.31·37-s + 16/7·49-s + 4.09·61-s + 6.55·73-s − 81-s − 4.06·97-s + 1.53·109-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(8430.11\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.856372879\)
\(L(\frac12)\) \(\approx\) \(4.856372879\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
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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

−7.00776086012816722965923251691, −6.76978532143042226733252310235, −6.55757142543924919475344300470, −6.25432680947147746309612162174, −6.24768064334008538503096196507, −5.76849217771663929498867791674, −5.57051271290936632965492023754, −5.48198343971334385019764684463, −5.20626979728758973302587263313, −5.03121135831868794505126131594, −4.85438075173400318834763113047, −4.27796491866144545188912360587, −4.01146614273138587152380748832, −3.84662769459138831875254747179, −3.80049165169194238548802076759, −3.73689878620226175120083138195, −3.08696525527140770893926355734, −2.88596053676174090621398366851, −2.76628894480770565724675399282, −2.03004059375596341958878570756, −2.00752560891054778093748432098, −1.87429084522762548668632244192, −1.02265928121029032497586413324, −0.953227584013924735430808775520, −0.57306524434504561419056415933, 0.57306524434504561419056415933, 0.953227584013924735430808775520, 1.02265928121029032497586413324, 1.87429084522762548668632244192, 2.00752560891054778093748432098, 2.03004059375596341958878570756, 2.76628894480770565724675399282, 2.88596053676174090621398366851, 3.08696525527140770893926355734, 3.73689878620226175120083138195, 3.80049165169194238548802076759, 3.84662769459138831875254747179, 4.01146614273138587152380748832, 4.27796491866144545188912360587, 4.85438075173400318834763113047, 5.03121135831868794505126131594, 5.20626979728758973302587263313, 5.48198343971334385019764684463, 5.57051271290936632965492023754, 5.76849217771663929498867791674, 6.24768064334008538503096196507, 6.25432680947147746309612162174, 6.55757142543924919475344300470, 6.76978532143042226733252310235, 7.00776086012816722965923251691

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