Properties

Label 8-1200e4-1.1-c2e4-0-18
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $1.14304\times 10^{6}$
Root an. cond. $5.71818$
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  − 6·9-s + 32·13-s − 48·17-s − 48·29-s − 80·37-s − 120·41-s + 52·49-s − 96·53-s + 56·61-s + 8·73-s + 27·81-s + 24·89-s + 56·97-s − 48·101-s + 280·109-s − 576·113-s − 192·117-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 288·153-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2/3·9-s + 2.46·13-s − 2.82·17-s − 1.65·29-s − 2.16·37-s − 2.92·41-s + 1.06·49-s − 1.81·53-s + 0.918·61-s + 8/73·73-s + 1/3·81-s + 0.269·89-s + 0.577·97-s − 0.475·101-s + 2.56·109-s − 5.09·113-s − 1.64·117-s + 2.80·121-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 + 1.88·153-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.14304\times 10^{6}\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.308431872\)
\(L(\frac12)\) \(\approx\) \(1.308431872\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
5 \( 1 \)
good7$D_4\times C_2$ \( 1 - 52 T^{2} + 2598 T^{4} - 52 p^{4} T^{6} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 - 340 T^{2} + 55302 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 - 16 T + 222 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 24 T + 542 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 868 T^{2} + 402918 T^{4} - 868 p^{4} T^{6} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 100 T^{2} + 377862 T^{4} - 100 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 + 24 T + 1646 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 964 T^{2} + 927366 T^{4} - 964 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 + 40 T + 1518 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{4} \)
43$D_4\times C_2$ \( 1 - 5380 T^{2} + 13889382 T^{4} - 5380 p^{4} T^{6} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 - 2212 T^{2} + 8033478 T^{4} - 2212 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 48 T + 4574 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 620 T^{2} + 23982342 T^{4} + 620 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 28 T + 1158 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 188 T^{2} - 19408602 T^{4} + 188 p^{4} T^{6} + p^{8} T^{8} \)
71$C_2^3$ \( 1 + 8060 T^{2} + 10662 p^{2} T^{4} + 8060 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 4 T + 9942 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 12868 T^{2} + 113720838 T^{4} - 12868 p^{4} T^{6} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 19780 T^{2} + 177798822 T^{4} - 19780 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 - 12 T + 15158 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{4} \)
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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

−6.80214274287827010795820581186, −6.66750936527277954504897570143, −6.24810246652016911930101309945, −6.23010327664442194205944015365, −5.81282475086683097820244105472, −5.67513229124836499782507366236, −5.51486749545344234279132264358, −5.31599861232935864517128127414, −4.75127238025683838965803392255, −4.74784196230258015341165947757, −4.64608477431252611883254491615, −4.13415676555331540296488734531, −4.01008728866875793926958782787, −3.60027561389014095386386829993, −3.41198100137483179611167033348, −3.37543022029526349019377703934, −3.18803154321025364964927329695, −2.45350053073465611441154772048, −2.35342785543453239058044400281, −2.03241615847625030520390729933, −1.79928678982369594079512036165, −1.35071205600456593276720958549, −1.28323575969235786946082450034, −0.41027379479192142026813010121, −0.25153955402928019706668075624, 0.25153955402928019706668075624, 0.41027379479192142026813010121, 1.28323575969235786946082450034, 1.35071205600456593276720958549, 1.79928678982369594079512036165, 2.03241615847625030520390729933, 2.35342785543453239058044400281, 2.45350053073465611441154772048, 3.18803154321025364964927329695, 3.37543022029526349019377703934, 3.41198100137483179611167033348, 3.60027561389014095386386829993, 4.01008728866875793926958782787, 4.13415676555331540296488734531, 4.64608477431252611883254491615, 4.74784196230258015341165947757, 4.75127238025683838965803392255, 5.31599861232935864517128127414, 5.51486749545344234279132264358, 5.67513229124836499782507366236, 5.81282475086683097820244105472, 6.23010327664442194205944015365, 6.24810246652016911930101309945, 6.66750936527277954504897570143, 6.80214274287827010795820581186

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