Properties

Label 8-1152e4-1.1-c2e4-0-10
Degree $8$
Conductor $1.761\times 10^{12}$
Sign $1$
Analytic cond. $970845.$
Root an. cond. $5.60265$
Motivic weight $2$
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  + 92·25-s + 4·49-s + 200·73-s − 760·97-s + 284·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-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  + 3.67·25-s + 4/49·49-s + 2.73·73-s − 7.83·97-s + 2.34·121-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 + 4·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(970845.\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1152} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.908684867\)
\(L(\frac12)\) \(\approx\) \(3.908684867\)
\(L(2)\) 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$C_2^2$ \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 10 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
17$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
19$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
29$C_2^2$ \( ( 1 + 818 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 38 T + p^{2} T^{2} )^{2}( 1 + 38 T + p^{2} T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
41$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
43$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
53$C_2^2$ \( ( 1 + 3218 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6862 T^{2} + p^{4} T^{4} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
67$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{4} \)
79$C_2$ \( ( 1 - 58 T + p^{2} T^{2} )^{2}( 1 + 58 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 4178 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 190 T + p^{2} T^{2} )^{4} \)
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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

−6.83483314747049142690532555878, −6.76512107375981661011391571533, −6.40256164978630995380455819596, −6.27504876914897286755346824105, −5.74194017524548546685894363839, −5.67290001916986564908025719962, −5.41407913558596952065603177629, −5.38668005720655873395109338967, −4.87615764422648190118350236222, −4.71481015480713285514728213170, −4.64071195967178653338179789815, −4.27265743869430506933979926611, −4.14826242169324120249194317948, −3.54006879705857753769415358668, −3.53043892537416800075037921730, −3.35846041890088462009654509504, −2.79078265767023872827905244865, −2.68470493628005419198723418462, −2.54688546269181576070764416388, −2.18684730535767663553017238104, −1.61123040821872616575888505994, −1.35677478127816702379097429599, −1.12355760716255419722170827842, −0.68079489459746825856764101310, −0.31453190218697586412276164792, 0.31453190218697586412276164792, 0.68079489459746825856764101310, 1.12355760716255419722170827842, 1.35677478127816702379097429599, 1.61123040821872616575888505994, 2.18684730535767663553017238104, 2.54688546269181576070764416388, 2.68470493628005419198723418462, 2.79078265767023872827905244865, 3.35846041890088462009654509504, 3.53043892537416800075037921730, 3.54006879705857753769415358668, 4.14826242169324120249194317948, 4.27265743869430506933979926611, 4.64071195967178653338179789815, 4.71481015480713285514728213170, 4.87615764422648190118350236222, 5.38668005720655873395109338967, 5.41407913558596952065603177629, 5.67290001916986564908025719962, 5.74194017524548546685894363839, 6.27504876914897286755346824105, 6.40256164978630995380455819596, 6.76512107375981661011391571533, 6.83483314747049142690532555878

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