Properties

Label 8-48e8-1.1-c2e4-0-11
Degree $8$
Conductor $2.818\times 10^{13}$
Sign $1$
Analytic cond. $1.55335\times 10^{7}$
Root an. cond. $7.92334$
Motivic weight $2$
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  − 96·25-s − 196·49-s + 384·73-s − 576·97-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 476·169-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 + ⋯
L(s)  = 1  − 3.83·25-s − 4·49-s + 5.26·73-s − 5.93·97-s − 4·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.81·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.55335\times 10^{7}\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7315148350\)
\(L(\frac12)\) \(\approx\) \(0.7315148350\)
\(L(2)\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 + 48 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
11$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
13$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 10 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + 480 T^{2} + p^{4} T^{4} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
29$C_2^2$ \( ( 1 + 1680 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
37$C_2$ \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + 1440 T^{2} + p^{4} T^{4} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
53$C_2^2$ \( ( 1 - 5040 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
61$C_2$ \( ( 1 - 120 T + p^{2} T^{2} )^{2}( 1 + 120 T + p^{2} T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2$ \( ( 1 - 96 T + p^{2} T^{2} )^{4} \)
79$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
89$C_2^2$ \( ( 1 + 12480 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 144 T + p^{2} T^{2} )^{4} \)
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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

−6.18218578877772472100235833310, −6.08263385305934633375010398408, −5.82287184795361258802538952150, −5.43256611014626679866277313907, −5.31985803890213966103242703215, −5.23448687442913221682203019954, −5.16711941158549230838030334824, −4.64865654372599771548220061868, −4.56352456407870083070616293950, −4.17156508503479391566961883679, −3.97562738636550127968618913005, −3.88345147126213382359122231364, −3.73368506844486739178244947320, −3.37290479873308110697482871740, −3.22188906562215573384401901285, −2.76277005879972610911122811248, −2.72425374233045724991112439856, −2.21143227908311375317370987045, −2.18194333067412668526138470071, −1.76567182193755470798264271207, −1.60670615511753637354622513128, −1.31365068187300220348025650684, −0.968329767209690045799321167933, −0.34607726208268549180599819722, −0.16319160761093751898636623169, 0.16319160761093751898636623169, 0.34607726208268549180599819722, 0.968329767209690045799321167933, 1.31365068187300220348025650684, 1.60670615511753637354622513128, 1.76567182193755470798264271207, 2.18194333067412668526138470071, 2.21143227908311375317370987045, 2.72425374233045724991112439856, 2.76277005879972610911122811248, 3.22188906562215573384401901285, 3.37290479873308110697482871740, 3.73368506844486739178244947320, 3.88345147126213382359122231364, 3.97562738636550127968618913005, 4.17156508503479391566961883679, 4.56352456407870083070616293950, 4.64865654372599771548220061868, 5.16711941158549230838030334824, 5.23448687442913221682203019954, 5.31985803890213966103242703215, 5.43256611014626679866277313907, 5.82287184795361258802538952150, 6.08263385305934633375010398408, 6.18218578877772472100235833310

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