Properties

Label 8-384e4-1.1-c6e4-0-1
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $6.09038\times 10^{7}$
Root an. cond. $9.39897$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 486·9-s − 1.15e4·17-s − 1.43e4·25-s + 7.49e4·41-s + 3.24e5·49-s + 2.51e6·73-s + 1.77e5·81-s − 5.50e6·89-s + 1.95e6·97-s − 6.32e6·113-s − 1.13e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 5.63e6·153-s + 157-s + 163-s + 167-s + 2.71e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s − 2.35·17-s − 0.917·25-s + 1.08·41-s + 2.75·49-s + 6.45·73-s + 1/3·81-s − 7.80·89-s + 2.14·97-s − 4.38·113-s − 0.640·121-s − 1.57·153-s + 0.563·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(6.09038\times 10^{7}\)
Root analytic conductor: \(9.39897\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3, 3, 3, 3 ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.325539905\)
\(L(\frac12)\) \(\approx\) \(2.325539905\)
\(L(4)\) 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$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 7166 T^{2} + p^{12} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 162290 T^{2} + p^{12} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 567074 T^{2} + p^{12} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 1359218 T^{2} + p^{12} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 2898 T + p^{6} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 1743550 T^{2} + p^{12} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 293443490 T^{2} + p^{12} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 976603426 T^{2} + p^{12} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1727677010 T^{2} + p^{12} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 4983392594 T^{2} + p^{12} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 18738 T + p^{6} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 5648607070 T^{2} + p^{12} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 1324300030 T^{2} + p^{12} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 23524698562 T^{2} + p^{12} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 26964818206 T^{2} + p^{12} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 8136149902 T^{2} + p^{12} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 78875702786 T^{2} + p^{12} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 79263457634 T^{2} + p^{12} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 8606 p T + p^{6} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 400981984274 T^{2} + p^{12} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 526419436610 T^{2} + p^{12} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 1375074 T + p^{6} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 489710 T + p^{6} T^{2} )^{4} \)
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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

−7.09043334192746213042732190944, −6.96611170853890438030759441486, −6.74290217924314945939109168217, −6.45794921491943076880096618464, −6.12679983564161528881908034032, −6.09200837984116699002760829127, −5.46593869072315604433969721156, −5.40609899904537487430425394022, −5.20248663862540180314850241707, −4.90992198146446574304214192404, −4.28314039892719423353218960884, −4.21204837772251925100569426271, −4.17506872885539156188605915225, −3.88583530045502840240003720090, −3.58851203622710696686237322064, −2.98500428587598290081915424481, −2.76460912106516714180348634485, −2.48133115201043464525156429490, −2.13810960387850130860096271744, −2.01966427262219881908699252172, −1.57187398046978677362870380447, −1.24665184994561214639361025379, −0.67177025891582614369999627356, −0.66696729893060139318968681303, −0.18294508809648738156515284914, 0.18294508809648738156515284914, 0.66696729893060139318968681303, 0.67177025891582614369999627356, 1.24665184994561214639361025379, 1.57187398046978677362870380447, 2.01966427262219881908699252172, 2.13810960387850130860096271744, 2.48133115201043464525156429490, 2.76460912106516714180348634485, 2.98500428587598290081915424481, 3.58851203622710696686237322064, 3.88583530045502840240003720090, 4.17506872885539156188605915225, 4.21204837772251925100569426271, 4.28314039892719423353218960884, 4.90992198146446574304214192404, 5.20248663862540180314850241707, 5.40609899904537487430425394022, 5.46593869072315604433969721156, 6.09200837984116699002760829127, 6.12679983564161528881908034032, 6.45794921491943076880096618464, 6.74290217924314945939109168217, 6.96611170853890438030759441486, 7.09043334192746213042732190944

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