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Properties

Label 8-42e8-1.1-c1e4-0-7
Degree $8$
Conductor $9.683\times 10^{12}$
Sign $1$
Analytic cond. $39364.3$
Root an. cond. $3.75308$
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  + 4·4-s + 12·16-s − 16·53-s + 32·64-s − 64·113-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s − 64·212-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 2·4-s + 3·16-s − 2.19·53-s + 4·64-s − 6.02·113-s + 4·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s − 4.39·212-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(39364.3\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.964180154\)
\(L(\frac12)\) \(\approx\) \(5.964180154\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_4\times C_2$ \( 1 + 48 T^{4} + p^{4} T^{8} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_4\times C_2$ \( 1 - 240 T^{4} + p^{4} T^{8} \)
17$C_4\times C_2$ \( 1 - 480 T^{4} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
41$C_4\times C_2$ \( 1 + 1440 T^{4} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_4\times C_2$ \( 1 - 2640 T^{4} + p^{4} T^{8} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_4\times C_2$ \( 1 - 10560 T^{4} + p^{4} T^{8} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_4\times C_2$ \( 1 + 12480 T^{4} + p^{4} T^{8} \)
97$C_4\times C_2$ \( 1 - 18720 T^{4} + p^{4} T^{8} \)
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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

−6.69787859028457075437143472755, −6.57851876510613705628372650611, −6.10910717576998323145674061108, −6.10321191807448411771875616533, −5.69507696326648819571209101051, −5.62022015370578142101922509291, −5.61366639427172163821020686088, −5.16485065376823062329750844185, −4.88312871357290656739655883051, −4.67044759101684378156081194681, −4.43719382837873118106220638101, −4.29822403628584107259314838131, −3.77269255955295260117279358492, −3.58928274412462435026470529822, −3.45123897326056742448175048326, −3.27540699370552373651557713586, −2.77277175510797967090037221933, −2.67868964341452706435772037562, −2.57401587983808503503574552960, −2.11611732093774316775310984575, −1.79381874791531551182182784847, −1.59798989481303707380366068421, −1.36438229032362288854306161651, −0.891598242377591644861462661617, −0.39604908726998785674236101829, 0.39604908726998785674236101829, 0.891598242377591644861462661617, 1.36438229032362288854306161651, 1.59798989481303707380366068421, 1.79381874791531551182182784847, 2.11611732093774316775310984575, 2.57401587983808503503574552960, 2.67868964341452706435772037562, 2.77277175510797967090037221933, 3.27540699370552373651557713586, 3.45123897326056742448175048326, 3.58928274412462435026470529822, 3.77269255955295260117279358492, 4.29822403628584107259314838131, 4.43719382837873118106220638101, 4.67044759101684378156081194681, 4.88312871357290656739655883051, 5.16485065376823062329750844185, 5.61366639427172163821020686088, 5.62022015370578142101922509291, 5.69507696326648819571209101051, 6.10321191807448411771875616533, 6.10910717576998323145674061108, 6.57851876510613705628372650611, 6.69787859028457075437143472755

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