Properties

Label 8-2352e4-1.1-c1e4-0-0
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $124410.$
Root an. cond. $4.33368$
Motivic weight $1$
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  + 2·9-s − 8·13-s + 16·25-s − 16·37-s + 24·61-s + 8·73-s − 5·81-s − 56·97-s + 8·109-s − 16·117-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2/3·9-s − 2.21·13-s + 16/5·25-s − 2.63·37-s + 3.07·61-s + 0.936·73-s − 5/9·81-s − 5.68·97-s + 0.766·109-s − 1.47·117-s − 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(124410.\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2138846338\)
\(L(\frac12)\) \(\approx\) \(0.2138846338\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{4} \)
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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

−6.28752490199887984473301779138, −6.27101526751309614562462925066, −6.11769899448783686891675218656, −5.65427546969965575706208879351, −5.28634358832790037908519631370, −5.18277385623651802495666436657, −5.16771072478647883491913355218, −5.00464343420923940090523869358, −4.89035482496582289109744010932, −4.57280219138456728305914531902, −4.14328079997480117371506743379, −3.95157822650082248186240223700, −3.93152627711715059196162055394, −3.69629734309122082441554295746, −3.20443249358079779490795880655, −2.94711158358216534346150022337, −2.93622319927902300706172653111, −2.39339501004370462916810343257, −2.34121596372163041027612478146, −2.33847562292306218577573793023, −1.60802948271258797810350980117, −1.30232165020607869713057838130, −1.28791014961678492540838418901, −0.73350901023005903068740718448, −0.081294069533011824491089266582, 0.081294069533011824491089266582, 0.73350901023005903068740718448, 1.28791014961678492540838418901, 1.30232165020607869713057838130, 1.60802948271258797810350980117, 2.33847562292306218577573793023, 2.34121596372163041027612478146, 2.39339501004370462916810343257, 2.93622319927902300706172653111, 2.94711158358216534346150022337, 3.20443249358079779490795880655, 3.69629734309122082441554295746, 3.93152627711715059196162055394, 3.95157822650082248186240223700, 4.14328079997480117371506743379, 4.57280219138456728305914531902, 4.89035482496582289109744010932, 5.00464343420923940090523869358, 5.16771072478647883491913355218, 5.18277385623651802495666436657, 5.28634358832790037908519631370, 5.65427546969965575706208879351, 6.11769899448783686891675218656, 6.27101526751309614562462925066, 6.28752490199887984473301779138

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