Properties

Label 8-12e12-1.1-c2e4-0-11
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $4.91490\times 10^{6}$
Root an. cond. $6.86182$
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  + 100·25-s − 46·49-s − 572·73-s − 676·97-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 382·169-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  + 4·25-s − 0.938·49-s − 7.83·73-s − 6.96·97-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 + 2.26·169-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(4.91490\times 10^{6}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.858590327\)
\(L(\frac12)\) \(\approx\) \(2.858590327\)
\(L(2)\) 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_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
7$C_2^2$ \( ( 1 + 23 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
13$C_2$ \( ( 1 - 23 T + p^{2} T^{2} )^{2}( 1 + 23 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 647 T^{2} + p^{4} T^{4} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
31$C_2^2$ \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 73 T + p^{2} T^{2} )^{2}( 1 + 73 T + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 3214 T^{2} + p^{4} T^{4} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
59$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
61$C_2$ \( ( 1 - 47 T + p^{2} T^{2} )^{2}( 1 + 47 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - 8809 T^{2} + p^{4} T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2$ \( ( 1 + 143 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 12361 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
89$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 169 T + p^{2} T^{2} )^{4} \)
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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

−6.52438360631731634831248051920, −6.39134088017906834703765649004, −5.92602105166253532227225383740, −5.70850621045482572438079032816, −5.47336863186252669336497189484, −5.46097881738878226716065924439, −5.35211363346596391887824857319, −4.75947693860258060159962465539, −4.55360638626132830451746097207, −4.51134352007357081452314562672, −4.48036516236638693475237473773, −4.07047438667192730896267213621, −3.77024396733366117563556880231, −3.48773447315695707029300942862, −3.14390510117531155552976869211, −2.92111238165941404057277869373, −2.77821517430590805278606673921, −2.59009259227525309597988721993, −2.51455465859533089390705836829, −1.59678704373156565034543998212, −1.56027971501526588203508044787, −1.30817808232301355385589108504, −1.22137844634960133945783234477, −0.46356727566563726786151498564, −0.27999664679558095872665227554, 0.27999664679558095872665227554, 0.46356727566563726786151498564, 1.22137844634960133945783234477, 1.30817808232301355385589108504, 1.56027971501526588203508044787, 1.59678704373156565034543998212, 2.51455465859533089390705836829, 2.59009259227525309597988721993, 2.77821517430590805278606673921, 2.92111238165941404057277869373, 3.14390510117531155552976869211, 3.48773447315695707029300942862, 3.77024396733366117563556880231, 4.07047438667192730896267213621, 4.48036516236638693475237473773, 4.51134352007357081452314562672, 4.55360638626132830451746097207, 4.75947693860258060159962465539, 5.35211363346596391887824857319, 5.46097881738878226716065924439, 5.47336863186252669336497189484, 5.70850621045482572438079032816, 5.92602105166253532227225383740, 6.39134088017906834703765649004, 6.52438360631731634831248051920

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