Properties

Label 8-1200e4-1.1-c1e4-0-16
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  + 44·31-s − 4·61-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  + 7.90·31-s − 0.512·61-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

\(L(1)\) \(\approx\) \(3.875592431\)
\(L(\frac12)\) \(\approx\) \(3.875592431\)
\(L(\frac{3}{2})\) 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$C_2^3$ \( 1 + 71 T^{4} + p^{4} T^{8} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2^3$ \( 1 + 191 T^{4} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
37$C_2^3$ \( 1 - 2062 T^{4} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^3$ \( 1 + 3191 T^{4} + p^{4} T^{8} \)
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$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
67$C_2^3$ \( 1 - 8809 T^{4} + p^{4} T^{8} \)
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$C_2^3$ \( 1 + 9071 T^{4} + p^{4} T^{8} \)
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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

−6.83731682203099200377387225402, −6.66939362729089187202259762112, −6.48149732173987194076299816332, −6.30234720244264038928220051807, −6.29249271334468575033423357659, −5.86250548661354191679816655373, −5.66163786826726662510357345434, −5.55004746568696187073607647491, −5.05845320738339636101057930467, −4.66070161048579647595081844975, −4.59098462127085621477486890515, −4.57912296300235672015326611217, −4.54908460597550949221496140204, −3.93053643832338263080004736896, −3.77890601369810442353567788891, −3.32019184603891760414973137728, −3.03586657001468690597935073141, −2.98666061868708644707400269449, −2.54625793442581065754182216381, −2.35031734485152074039002646127, −2.25256315194500926898491817944, −1.42920356221139762698288822257, −1.21062990634165342564177201413, −0.949317840475400380255940887836, −0.48230464576594586914570394977, 0.48230464576594586914570394977, 0.949317840475400380255940887836, 1.21062990634165342564177201413, 1.42920356221139762698288822257, 2.25256315194500926898491817944, 2.35031734485152074039002646127, 2.54625793442581065754182216381, 2.98666061868708644707400269449, 3.03586657001468690597935073141, 3.32019184603891760414973137728, 3.77890601369810442353567788891, 3.93053643832338263080004736896, 4.54908460597550949221496140204, 4.57912296300235672015326611217, 4.59098462127085621477486890515, 4.66070161048579647595081844975, 5.05845320738339636101057930467, 5.55004746568696187073607647491, 5.66163786826726662510357345434, 5.86250548661354191679816655373, 6.29249271334468575033423357659, 6.30234720244264038928220051807, 6.48149732173987194076299816332, 6.66939362729089187202259762112, 6.83731682203099200377387225402

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