Properties

Label 8-1050e4-1.1-c1e4-0-17
Degree $8$
Conductor $1.216\times 10^{12}$
Sign $1$
Analytic cond. $4941.57$
Root an. cond. $2.89556$
Motivic weight $1$
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  + 4-s + 9-s − 10·11-s + 16·19-s + 20·29-s − 6·31-s + 36-s − 10·44-s + 13·49-s − 22·59-s + 12·61-s − 64-s + 8·71-s + 16·76-s + 6·79-s − 12·89-s − 10·99-s − 20·101-s − 4·109-s + 20·116-s + 47·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s − 3.01·11-s + 3.67·19-s + 3.71·29-s − 1.07·31-s + 1/6·36-s − 1.50·44-s + 13/7·49-s − 2.86·59-s + 1.53·61-s − 1/8·64-s + 0.949·71-s + 1.83·76-s + 0.675·79-s − 1.27·89-s − 1.00·99-s − 1.99·101-s − 0.383·109-s + 1.85·116-s + 4.27·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-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: $1$
Analytic conductor: \(4941.57\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
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

\(L(1)\) \(\approx\) \(3.830993695\)
\(L(\frac12)\) \(\approx\) \(3.830993695\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
good11$C_2^2$ \( ( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2^3$ \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \)
23$C_2^3$ \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
73$C_2^2$$\times$$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.25922150628742861346754953079, −7.09203633572202663977570051210, −6.52152700919763019293472512045, −6.49142901847035075618646334710, −6.33117746254756987682754920623, −5.83879135093137605105981147669, −5.62078783408022173939881494122, −5.45291232493097690878108713892, −5.38400007955761896827362294891, −4.92479944315039988266046788853, −4.92436881377592436431929597252, −4.83927145360409769722314477828, −4.29009493202707823399939619761, −4.10041992755618532365209584032, −3.71571472285261648147546175966, −3.39961154514182652700317008574, −2.99125595109485557554377672783, −2.95418766513610313660702410090, −2.73979647377619173140643440214, −2.54699166389504677205237037881, −2.20006788793276806493644399026, −1.54356944354942752401375049079, −1.42711203661123366459042099296, −0.75164977503568057511418596801, −0.57234456117050874678171243445, 0.57234456117050874678171243445, 0.75164977503568057511418596801, 1.42711203661123366459042099296, 1.54356944354942752401375049079, 2.20006788793276806493644399026, 2.54699166389504677205237037881, 2.73979647377619173140643440214, 2.95418766513610313660702410090, 2.99125595109485557554377672783, 3.39961154514182652700317008574, 3.71571472285261648147546175966, 4.10041992755618532365209584032, 4.29009493202707823399939619761, 4.83927145360409769722314477828, 4.92436881377592436431929597252, 4.92479944315039988266046788853, 5.38400007955761896827362294891, 5.45291232493097690878108713892, 5.62078783408022173939881494122, 5.83879135093137605105981147669, 6.33117746254756987682754920623, 6.49142901847035075618646334710, 6.52152700919763019293472512045, 7.09203633572202663977570051210, 7.25922150628742861346754953079

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