Properties

Label 8-1200e4-1.1-c1e4-0-9
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  − 28·31-s − 52·61-s − 9·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  − 5.02·31-s − 6.65·61-s − 81-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 + 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 + 0.0655·233-s + 0.0646·239-s + 0.0644·241-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: induced by $\chi_{1200} (1, \cdot )$
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

\(L(1)\) \(\approx\) \(1.124306659\)
\(L(\frac12)\) \(\approx\) \(1.124306659\)
\(L(\frac{3}{2})\) 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$C_2^3$ \( 1 + 23 T^{4} + p^{4} T^{8} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2^3$ \( 1 - 337 T^{4} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
37$C_2^3$ \( 1 - 2062 T^{4} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^3$ \( 1 + 23 T^{4} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{4} \)
67$C_2^3$ \( 1 + 2903 T^{4} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2^3$ \( 1 - 8542 T^{4} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^3$ \( 1 + 9743 T^{4} + p^{4} T^{8} \)
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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

−7.08787575767419269011537394083, −6.69795574940877162864770533604, −6.65373616702408974849958648322, −6.25905167090188330134935932664, −5.89721627387279533303010670683, −5.85405537714350411410635522330, −5.76845981431194705053831199387, −5.45370120231366968070528596503, −5.22866315630624823397755240879, −4.92833393825861352073464548962, −4.69769972474846764067093226777, −4.36014247350366254408309390724, −4.33012882558652088259908790023, −3.86750781616940135731603473403, −3.81736524990983242471248363028, −3.33454330567655595976170662630, −3.11498564301799755441079090310, −2.95280983750729495426076496258, −2.87485216423345783022698885508, −2.05968572713021108732908885601, −1.77087649217755426898903470981, −1.71562304189961200071544134476, −1.66488608485280322720609565840, −0.72206556061217361207384685489, −0.26700128743525946645959404261, 0.26700128743525946645959404261, 0.72206556061217361207384685489, 1.66488608485280322720609565840, 1.71562304189961200071544134476, 1.77087649217755426898903470981, 2.05968572713021108732908885601, 2.87485216423345783022698885508, 2.95280983750729495426076496258, 3.11498564301799755441079090310, 3.33454330567655595976170662630, 3.81736524990983242471248363028, 3.86750781616940135731603473403, 4.33012882558652088259908790023, 4.36014247350366254408309390724, 4.69769972474846764067093226777, 4.92833393825861352073464548962, 5.22866315630624823397755240879, 5.45370120231366968070528596503, 5.76845981431194705053831199387, 5.85405537714350411410635522330, 5.89721627387279533303010670683, 6.25905167090188330134935932664, 6.65373616702408974849958648322, 6.69795574940877162864770533604, 7.08787575767419269011537394083

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