Properties

Label 8-432e4-1.1-c5e4-0-0
Degree $8$
Conductor $34828517376$
Sign $1$
Analytic cond. $2.30450\times 10^{7}$
Root an. cond. $8.32380$
Motivic weight $5$
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  − 2.51e3·13-s + 1.26e3·25-s + 3.34e4·37-s + 3.86e4·49-s + 6.71e4·61-s − 2.15e5·73-s − 6.25e4·97-s + 5.97e5·109-s − 3.40e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.47e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 4.12·13-s + 0.405·25-s + 4.01·37-s + 2.30·49-s + 2.31·61-s − 4.73·73-s − 0.674·97-s + 4.82·109-s − 2.11·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 6.65·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(2.30450\times 10^{7}\)
Root analytic conductor: \(8.32380\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{12} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.06357347745\)
\(L(\frac12)\) \(\approx\) \(0.06357347745\)
\(L(\frac{7}{2})\) 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$C_2^2$ \( ( 1 - 634 T^{2} + p^{10} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 19331 T^{2} + p^{10} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 170470 T^{2} + p^{10} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 629 T + p^{5} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 2699314 T^{2} + p^{10} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 4808315 T^{2} + p^{10} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 5474786 T^{2} + p^{10} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 34530202 T^{2} + p^{10} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 34702610 T^{2} + p^{10} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 8363 T + p^{5} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 207249106 T^{2} + p^{10} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 203991434 T^{2} + p^{10} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 178322446 T^{2} + p^{10} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 765045110 T^{2} + p^{10} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 1174955206 T^{2} + p^{10} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 16799 T + p^{5} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 925982653 T^{2} + p^{10} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 1525194290 T^{2} + p^{10} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 53849 T + p^{5} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 5394251123 T^{2} + p^{10} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 4092740038 T^{2} + p^{10} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 1892591102 T^{2} + p^{10} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 15629 T + p^{5} T^{2} )^{4} \)
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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

−7.39210961554400598393642601265, −7.06620614646052018570375449867, −7.04682221852144401876472788208, −6.27657795733835911686663968769, −6.25413936934416636511157479342, −6.09553556515394526803358765175, −5.57528806570239915904200647023, −5.54706796682710178208941105295, −5.11897824623329929186109205506, −4.82668429549984592128684350405, −4.63616762402701208631465286501, −4.57149883324295608809306743141, −4.17404625924925836700667624908, −3.90171550679689877275597595623, −3.58454860854183811654477080736, −3.00086371170668247068939528943, −2.71352937169203417804904498707, −2.59999867647799049679859689799, −2.31314617114203597449685238070, −2.29355445679787590242137405991, −1.67027151263070487274985811365, −1.09276329001893809499609862159, −0.953263820100425704452815425009, −0.50421981914375098150864144301, −0.03678291247804213795371778794, 0.03678291247804213795371778794, 0.50421981914375098150864144301, 0.953263820100425704452815425009, 1.09276329001893809499609862159, 1.67027151263070487274985811365, 2.29355445679787590242137405991, 2.31314617114203597449685238070, 2.59999867647799049679859689799, 2.71352937169203417804904498707, 3.00086371170668247068939528943, 3.58454860854183811654477080736, 3.90171550679689877275597595623, 4.17404625924925836700667624908, 4.57149883324295608809306743141, 4.63616762402701208631465286501, 4.82668429549984592128684350405, 5.11897824623329929186109205506, 5.54706796682710178208941105295, 5.57528806570239915904200647023, 6.09553556515394526803358765175, 6.25413936934416636511157479342, 6.27657795733835911686663968769, 7.04682221852144401876472788208, 7.06620614646052018570375449867, 7.39210961554400598393642601265

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