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  − 2·3-s − 2·4-s − 6·7-s + 2·9-s + 4·12-s + 3·16-s + 12·17-s + 12·21-s − 6·27-s + 12·28-s − 4·36-s − 12·37-s − 8·41-s − 16·43-s − 8·47-s − 6·48-s + 18·49-s − 24·51-s − 12·63-s − 4·64-s + 48·67-s − 24·68-s + 11·81-s − 4·83-s − 24·84-s + 40·89-s + 32·101-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 2.26·7-s + 2/3·9-s + 1.15·12-s + 3/4·16-s + 2.91·17-s + 2.61·21-s − 1.15·27-s + 2.26·28-s − 2/3·36-s − 1.97·37-s − 1.24·41-s − 2.43·43-s − 1.16·47-s − 0.866·48-s + 18/7·49-s − 3.36·51-s − 1.51·63-s − 1/2·64-s + 5.86·67-s − 2.91·68-s + 11/9·81-s − 0.439·83-s − 2.61·84-s + 4.23·89-s + 3.18·101-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

\(L(1)\) \(\approx\) \(0.9581604347\)
\(L(\frac12)\) \(\approx\) \(0.9581604347\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
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$D_4\times C_2$ \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_4\times C_2$ \( 1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
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$D_{4}$ \( ( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
71$D_4\times C_2$ \( 1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 + 2 T - 78 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.07695037548084124573798084643, −6.67117526894759038248053968955, −6.65203393111786112308311253682, −6.50770455237962211578120041866, −6.24166989447364325111266530492, −5.94551444455001453471782069814, −5.54087752498860918235353038522, −5.51249551236562110079772572073, −5.32513458168058395407693369992, −5.26672732927500359932293631356, −4.95237862709696937451527161312, −4.48802128430245909023111308315, −4.46366704691807538583500323137, −3.97914396270348012768379419976, −3.57024867946714134826244869177, −3.40679157353147941679105701482, −3.33038250647856729577735503743, −3.21877736379699181694207944397, −3.12933767218580079014471923568, −2.05948800320747712998092132007, −2.02127906889695795207952231546, −1.80681704106805753861817167884, −0.967212131700865204571020436913, −0.57542410400066385690911800806, −0.49127357336985272608076892653, 0.49127357336985272608076892653, 0.57542410400066385690911800806, 0.967212131700865204571020436913, 1.80681704106805753861817167884, 2.02127906889695795207952231546, 2.05948800320747712998092132007, 3.12933767218580079014471923568, 3.21877736379699181694207944397, 3.33038250647856729577735503743, 3.40679157353147941679105701482, 3.57024867946714134826244869177, 3.97914396270348012768379419976, 4.46366704691807538583500323137, 4.48802128430245909023111308315, 4.95237862709696937451527161312, 5.26672732927500359932293631356, 5.32513458168058395407693369992, 5.51249551236562110079772572073, 5.54087752498860918235353038522, 5.94551444455001453471782069814, 6.24166989447364325111266530492, 6.50770455237962211578120041866, 6.65203393111786112308311253682, 6.67117526894759038248053968955, 7.07695037548084124573798084643

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