Properties

Label 8-384e4-1.1-c3e4-0-1
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $263505.$
Root an. cond. $4.75990$
Motivic weight $3$
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  − 54·9-s + 284·25-s + 940·49-s − 1.28e3·73-s + 2.18e3·81-s − 2.29e3·97-s − 5.26e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 1.53e4·225-s + ⋯
L(s)  = 1  − 2·9-s + 2.27·25-s + 2.74·49-s − 2.06·73-s + 3·81-s − 2.40·97-s − 3.95·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s − 4.54·225-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(263505.\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.1831540006\)
\(L(\frac12)\) \(\approx\) \(0.1831540006\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 142 T^{2} + p^{6} T^{4} )^{2} \)
7$C_2$ \( ( 1 - 34 T + p^{3} T^{2} )^{2}( 1 + 34 T + p^{3} T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + 2630 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
17$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
19$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 1150 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 70 T + p^{3} T^{2} )^{2}( 1 + 70 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
41$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
43$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 38446 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 103430 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
67$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
73$C_2$ \( ( 1 + 322 T + p^{3} T^{2} )^{4} \)
79$C_2$ \( ( 1 - 1370 T + p^{3} T^{2} )^{2}( 1 + 1370 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( ( 1 - 363274 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 574 T + p^{3} T^{2} )^{4} \)
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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

−7.894089727863547590176517569708, −7.45197788894682430086376673578, −7.20962577963725212007060827073, −7.16315580762491008990511177209, −6.69330962535993420628985677129, −6.45097038064977193442317109632, −6.29677580421612525267113647537, −6.03262028679669973939631825311, −5.56883377452227372291410318242, −5.40717127083419086611169310767, −5.31446810414865003422545089125, −5.09097511971335243377465277475, −4.46394019744510504276887378114, −4.44256206640290194843437827735, −3.90623325847472728747429711299, −3.84726455468621692537444513998, −3.14810597289507400135949194956, −3.01678422283069300545319096928, −2.83984442456675512664922667349, −2.46469733520374355879346299386, −2.16391865170383320522150955332, −1.58140545468354759443746766838, −0.992316993639454409652799630127, −0.857872735649538423095666894694, −0.07475387363303661820435515626, 0.07475387363303661820435515626, 0.857872735649538423095666894694, 0.992316993639454409652799630127, 1.58140545468354759443746766838, 2.16391865170383320522150955332, 2.46469733520374355879346299386, 2.83984442456675512664922667349, 3.01678422283069300545319096928, 3.14810597289507400135949194956, 3.84726455468621692537444513998, 3.90623325847472728747429711299, 4.44256206640290194843437827735, 4.46394019744510504276887378114, 5.09097511971335243377465277475, 5.31446810414865003422545089125, 5.40717127083419086611169310767, 5.56883377452227372291410318242, 6.03262028679669973939631825311, 6.29677580421612525267113647537, 6.45097038064977193442317109632, 6.69330962535993420628985677129, 7.16315580762491008990511177209, 7.20962577963725212007060827073, 7.45197788894682430086376673578, 7.894089727863547590176517569708

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