Properties

Label 8-3200e4-1.1-c1e4-0-30
Degree $8$
Conductor $1.049\times 10^{14}$
Sign $1$
Analytic cond. $426293.$
Root an. cond. $5.05491$
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  + 8·9-s + 48·41-s − 8·49-s + 30·81-s + 24·89-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 8/3·9-s + 7.49·41-s − 8/7·49-s + 10/3·81-s + 2.54·89-s + 4·121-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 + 4·169-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(426293.\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.49602111\)
\(L(\frac12)\) \(\approx\) \(10.49602111\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2$ \( ( 1 - p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−6.12505977693934690802267176816, −5.86197909919710251196310555134, −5.66269012908206579785187372778, −5.65506345717924564612781294588, −5.55258207921656896071548240671, −4.85704260264121078331203349494, −4.85377751630285356745847945586, −4.65937574001803619265214304332, −4.42383587629519815647301654860, −4.30107082651958227584619783553, −4.20799995469679379846350094327, −4.01821367015639507002774943088, −3.53110413824044632371810507149, −3.52198262449594226102679596540, −3.32575124432422205203599727838, −2.85244636023037056017606456719, −2.61432698199595849931782372763, −2.39281445748718419741787766846, −2.25117285212118282939805919152, −1.85433128560553125211803212941, −1.68515344867657993320041582016, −1.26837630764673246398586566190, −1.00821793580506145359155993651, −0.70032565796597816919323556420, −0.57913438628799109247460072118, 0.57913438628799109247460072118, 0.70032565796597816919323556420, 1.00821793580506145359155993651, 1.26837630764673246398586566190, 1.68515344867657993320041582016, 1.85433128560553125211803212941, 2.25117285212118282939805919152, 2.39281445748718419741787766846, 2.61432698199595849931782372763, 2.85244636023037056017606456719, 3.32575124432422205203599727838, 3.52198262449594226102679596540, 3.53110413824044632371810507149, 4.01821367015639507002774943088, 4.20799995469679379846350094327, 4.30107082651958227584619783553, 4.42383587629519815647301654860, 4.65937574001803619265214304332, 4.85377751630285356745847945586, 4.85704260264121078331203349494, 5.55258207921656896071548240671, 5.65506345717924564612781294588, 5.66269012908206579785187372778, 5.86197909919710251196310555134, 6.12505977693934690802267176816

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