Properties

Label 8-384e4-1.1-c2e4-0-0
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $11985.7$
Root an. cond. $3.23469$
Motivic weight $2$
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  − 16·5-s + 2·9-s + 60·25-s − 208·29-s − 32·45-s − 36·49-s − 80·53-s − 200·73-s − 77·81-s + 200·97-s + 368·101-s − 444·121-s + 720·125-s + 127-s + 131-s + 137-s + 139-s + 3.32e3·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 3.19·5-s + 2/9·9-s + 12/5·25-s − 7.17·29-s − 0.711·45-s − 0.734·49-s − 1.50·53-s − 2.73·73-s − 0.950·81-s + 2.06·97-s + 3.64·101-s − 3.66·121-s + 5.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 22.9·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.213·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(11985.7\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.06380874251\)
\(L(\frac12)\) \(\approx\) \(0.06380874251\)
\(L(2)\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{4} T^{4} \)
good5$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{4} \)
7$C_2^2$ \( ( 1 + 18 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 222 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 18 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 258 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 322 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 52 T + p^{2} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 1202 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 142 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2082 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2402 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 322 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2$ \( ( 1 + 20 T + p^{2} T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - 3618 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 7122 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 7042 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 3682 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 6002 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 3198 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 10078 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{4} \)
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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

−7.75166623788681679248650177751, −7.71835849348342913367122227067, −7.71089317366716823868673789761, −7.38292563030178520267310817739, −7.33118994576063656333441502810, −6.81399737835027693146003371190, −6.71020393402280904749582006875, −6.04363436313573710962835260513, −5.84390025854139771137997729019, −5.79440845256930210629715841074, −5.48568703416835118046212706929, −5.07387853715085123425158374915, −4.66955066333141990014183063718, −4.44477026422559753973472084666, −4.09646598894109535817606068237, −3.81165165516651943210809014544, −3.79397000014203190252817605218, −3.42638777557875419206071808307, −3.38298397368546402677643625056, −2.80946948900710314451898410473, −1.95017325878480092345137962446, −1.91729066472765812766389289216, −1.58883228402973684387131755115, −0.44329692545479391357174551610, −0.10934780135488740737296090459, 0.10934780135488740737296090459, 0.44329692545479391357174551610, 1.58883228402973684387131755115, 1.91729066472765812766389289216, 1.95017325878480092345137962446, 2.80946948900710314451898410473, 3.38298397368546402677643625056, 3.42638777557875419206071808307, 3.79397000014203190252817605218, 3.81165165516651943210809014544, 4.09646598894109535817606068237, 4.44477026422559753973472084666, 4.66955066333141990014183063718, 5.07387853715085123425158374915, 5.48568703416835118046212706929, 5.79440845256930210629715841074, 5.84390025854139771137997729019, 6.04363436313573710962835260513, 6.71020393402280904749582006875, 6.81399737835027693146003371190, 7.33118994576063656333441502810, 7.38292563030178520267310817739, 7.71089317366716823868673789761, 7.71835849348342913367122227067, 7.75166623788681679248650177751

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