Properties

Label 8-384e4-1.1-c4e4-0-1
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $2.48257\times 10^{6}$
Root an. cond. $6.30032$
Motivic weight $4$
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  + 54·9-s + 440·17-s + 2.21e3·25-s + 5.12e3·41-s + 1.82e3·49-s − 1.26e4·73-s + 2.18e3·81-s + 6.20e3·89-s − 3.20e4·97-s + 4.31e4·113-s − 5.77e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.37e4·153-s + 157-s + 163-s + 167-s − 2.51e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s + 1.52·17-s + 3.53·25-s + 3.05·41-s + 0.761·49-s − 2.37·73-s + 1/3·81-s + 0.782·89-s − 3.40·97-s + 3.37·113-s − 3.94·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 1.01·153-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 0.880·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.48257\times 10^{6}\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(5.109649742\)
\(L(\frac12)\) \(\approx\) \(5.109649742\)
\(L(3)\) 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$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 1106 T^{2} + p^{8} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 914 T^{2} + p^{8} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 28850 T^{2} + p^{8} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 12574 T^{2} + p^{8} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 110 T + p^{4} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 249842 T^{2} + p^{8} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 544130 T^{2} + p^{8} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1362578 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 209710 T^{2} + p^{8} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 1810658 T^{2} + p^{8} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 1282 T + p^{4} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 512690 T^{2} + p^{8} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 2900930 T^{2} + p^{8} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 4469038 T^{2} + p^{8} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 15952078 T^{2} + p^{8} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 26770082 T^{2} + p^{8} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 29653874 T^{2} + p^{8} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 13483010 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 3170 T + p^{4} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 75470162 T^{2} + p^{8} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 75317234 T^{2} + p^{8} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 1550 T + p^{4} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 8018 T + p^{4} T^{2} )^{4} \)
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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

−7.52430735458913249006885169643, −7.19217945210221596688079980888, −7.17339986586596698173348790766, −7.02841747537431841092943872961, −6.39609994691329036861855829779, −6.37471614800059391636846938092, −6.12238274392641104379948463586, −5.69017508859430772656749566244, −5.46799402504972367312424667300, −5.30714586429216543879901317480, −4.99955013969942882618362645620, −4.43062190233919993747276681173, −4.39447887810182400534859530290, −4.39060746429833333398829066974, −3.76626348837814438112316804984, −3.40876379731055636672971212969, −3.16159830804685751362627477281, −2.89214420216272058261810999628, −2.47850234505736950985189126087, −2.37922208328600758689166797373, −1.69283079674139068853322708992, −1.22097985485714397922002918178, −0.971995236939304971463894420807, −0.967024334616953900884399494641, −0.28338875003646561390939204738, 0.28338875003646561390939204738, 0.967024334616953900884399494641, 0.971995236939304971463894420807, 1.22097985485714397922002918178, 1.69283079674139068853322708992, 2.37922208328600758689166797373, 2.47850234505736950985189126087, 2.89214420216272058261810999628, 3.16159830804685751362627477281, 3.40876379731055636672971212969, 3.76626348837814438112316804984, 4.39060746429833333398829066974, 4.39447887810182400534859530290, 4.43062190233919993747276681173, 4.99955013969942882618362645620, 5.30714586429216543879901317480, 5.46799402504972367312424667300, 5.69017508859430772656749566244, 6.12238274392641104379948463586, 6.37471614800059391636846938092, 6.39609994691329036861855829779, 7.02841747537431841092943872961, 7.17339986586596698173348790766, 7.19217945210221596688079980888, 7.52430735458913249006885169643

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