Properties

Label 8-966e4-1.1-c1e4-0-9
Degree $8$
Conductor $870780120336$
Sign $1$
Analytic cond. $3540.11$
Root an. cond. $2.77732$
Motivic weight $1$
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  + 6·3-s + 4-s − 10·7-s + 21·9-s + 6·12-s + 24·19-s − 60·21-s − 2·25-s + 54·27-s − 10·28-s + 12·31-s + 21·36-s + 4·37-s + 40·43-s + 61·49-s + 144·57-s − 48·61-s − 210·63-s − 64-s + 8·67-s + 30·73-s − 12·75-s + 24·76-s − 22·79-s + 108·81-s − 60·84-s + 72·93-s + ⋯
L(s)  = 1  + 3.46·3-s + 1/2·4-s − 3.77·7-s + 7·9-s + 1.73·12-s + 5.50·19-s − 13.0·21-s − 2/5·25-s + 10.3·27-s − 1.88·28-s + 2.15·31-s + 7/2·36-s + 0.657·37-s + 6.09·43-s + 61/7·49-s + 19.0·57-s − 6.14·61-s − 26.4·63-s − 1/8·64-s + 0.977·67-s + 3.51·73-s − 1.38·75-s + 2.75·76-s − 2.47·79-s + 12·81-s − 6.54·84-s + 7.46·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(3540.11\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(15.34398765\)
\(L(\frac12)\) \(\approx\) \(15.34398765\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
good5$C_2^3$ \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^3$ \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 31 T^{2} + 672 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 - 19 T^{2} - 1848 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^3$ \( 1 + 74 T^{2} + 1995 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 130 T^{2} + 8979 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.38024207767404975473705119753, −7.10113853968234431828198042677, −6.90604776026748940453043342288, −6.74322152754312084948694677583, −6.27069278489263540851888730706, −5.98566536918012209530376388698, −5.89447425439026169374913450361, −5.77893279322580307327085732682, −5.66176365005791203957000786204, −4.82840659399795029565920689336, −4.63125956071794473475543601928, −4.62873011110366232595882637017, −4.07750160909396446809897453633, −3.80610578686650108355649272186, −3.49900781072499852421896034421, −3.35813316044301760055809008869, −3.34615779536844302767168017299, −3.01422550492283049541736641957, −2.72758457602605648732733635639, −2.60164062846492449596037687328, −2.51616565001459956290848056573, −2.10168030921113068437880000780, −1.17874714926869710279135177254, −1.10342726057049501788041806710, −0.807660225016461357025373535174, 0.807660225016461357025373535174, 1.10342726057049501788041806710, 1.17874714926869710279135177254, 2.10168030921113068437880000780, 2.51616565001459956290848056573, 2.60164062846492449596037687328, 2.72758457602605648732733635639, 3.01422550492283049541736641957, 3.34615779536844302767168017299, 3.35813316044301760055809008869, 3.49900781072499852421896034421, 3.80610578686650108355649272186, 4.07750160909396446809897453633, 4.62873011110366232595882637017, 4.63125956071794473475543601928, 4.82840659399795029565920689336, 5.66176365005791203957000786204, 5.77893279322580307327085732682, 5.89447425439026169374913450361, 5.98566536918012209530376388698, 6.27069278489263540851888730706, 6.74322152754312084948694677583, 6.90604776026748940453043342288, 7.10113853968234431828198042677, 7.38024207767404975473705119753

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