Properties

Degree $8$
Conductor $1.216\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 2·3-s + 10·4-s − 8·6-s + 6·7-s + 20·8-s + 2·9-s − 20·12-s − 4·13-s + 24·14-s + 35·16-s + 8·18-s − 12·21-s + 16·23-s − 40·24-s − 16·26-s − 6·27-s + 60·28-s + 56·32-s + 20·36-s + 8·39-s + 8·41-s − 48·42-s + 64·46-s − 70·48-s + 18·49-s − 40·52-s + ⋯
L(s)  = 1  + 2.82·2-s − 1.15·3-s + 5·4-s − 3.26·6-s + 2.26·7-s + 7.07·8-s + 2/3·9-s − 5.77·12-s − 1.10·13-s + 6.41·14-s + 35/4·16-s + 1.88·18-s − 2.61·21-s + 3.33·23-s − 8.16·24-s − 3.13·26-s − 1.15·27-s + 11.3·28-s + 9.89·32-s + 10/3·36-s + 1.28·39-s + 1.24·41-s − 7.40·42-s + 9.43·46-s − 10.1·48-s + 18/7·49-s − 5.54·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(28.98435315\)
\(L(\frac12)\) \(\approx\) \(28.98435315\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$D_{4}$ \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 40 T^{2} + 798 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
19$C_4\times C_2$ \( 1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
29$D_4\times C_2$ \( 1 - 24 T^{2} + 1646 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 64 T^{2} + 2446 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 120 T^{2} + 6158 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 20 T^{2} - 1322 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 140 T^{2} + 8998 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 184 T^{2} + 15406 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 160 T^{2} + 19198 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 6 T + 198 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−7.06110224803891365875955789111, −6.76708913248062965026807378408, −6.70138906176329705354920358828, −6.19351256455667677641751673872, −6.09766930998894013556334516414, −5.86276503884085867126285581071, −5.51207586273090543294886826879, −5.27831764470113514120003743645, −5.20569160946380895495173071772, −5.07615583130982832862373006489, −4.92613237736275825366379992445, −4.65329896354591816174690246371, −4.51672718268051537116152611533, −4.13897608242807353134322195402, −3.89689964513329360040216535979, −3.75557977220978641492751469482, −3.26890809992590836219366957442, −3.21633726460588618721200352800, −2.52919806393723827691262263543, −2.45181275391618129208314372198, −2.31363722480896651569688592899, −1.92054078373990136184851912075, −1.45036470134578584843213020945, −0.980090286496742946707337151453, −0.869774493084748360691975202743, 0.869774493084748360691975202743, 0.980090286496742946707337151453, 1.45036470134578584843213020945, 1.92054078373990136184851912075, 2.31363722480896651569688592899, 2.45181275391618129208314372198, 2.52919806393723827691262263543, 3.21633726460588618721200352800, 3.26890809992590836219366957442, 3.75557977220978641492751469482, 3.89689964513329360040216535979, 4.13897608242807353134322195402, 4.51672718268051537116152611533, 4.65329896354591816174690246371, 4.92613237736275825366379992445, 5.07615583130982832862373006489, 5.20569160946380895495173071772, 5.27831764470113514120003743645, 5.51207586273090543294886826879, 5.86276503884085867126285581071, 6.09766930998894013556334516414, 6.19351256455667677641751673872, 6.70138906176329705354920358828, 6.76708913248062965026807378408, 7.06110224803891365875955789111

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