Properties

Label 8-50e8-1.1-c0e4-0-2
Degree $8$
Conductor $3.906\times 10^{13}$
Sign $1$
Analytic cond. $2.42319$
Root an. cond. $1.11698$
Motivic weight $0$
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  + 2-s − 3·3-s − 3·6-s + 2·7-s + 6·9-s + 2·14-s + 6·18-s − 6·21-s − 3·23-s − 10·27-s + 3·29-s − 32-s − 2·41-s − 6·42-s + 2·43-s − 3·46-s − 3·47-s + 49-s − 10·54-s + 3·58-s + 3·61-s + 12·63-s − 64-s + 2·67-s + 9·69-s + 15·81-s − 2·82-s + ⋯
L(s)  = 1  + 2-s − 3·3-s − 3·6-s + 2·7-s + 6·9-s + 2·14-s + 6·18-s − 6·21-s − 3·23-s − 10·27-s + 3·29-s − 32-s − 2·41-s − 6·42-s + 2·43-s − 3·46-s − 3·47-s + 49-s − 10·54-s + 3·58-s + 3·61-s + 12·63-s − 64-s + 2·67-s + 9·69-s + 15·81-s − 2·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(2.42319\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2500} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{16} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.011234410\)
\(L(\frac12)\) \(\approx\) \(1.011234410\)
\(L(1)\) 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$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
5 \( 1 \)
good3$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
7$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
23$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
29$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
47$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
89$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−6.53205321172626881922714777426, −6.17404689567621861619154803652, −6.07971831909751252502381221942, −5.98130236904461519546577517603, −5.50004580793099174433454266777, −5.41147387494631502569469772070, −5.24759176283247482483942251059, −5.00487188715650907674742067289, −4.92010890859246411725800940399, −4.78830838677638834278201643373, −4.50343880762503994557043533552, −4.42710052881032456329789226494, −4.30111191556061734465061944626, −3.81165193560217516876460988218, −3.62073683677735157437505673779, −3.54479439617205875246132175120, −3.52081820727396470738447883201, −2.54323362755843291156891669940, −2.41618162202106074767020392860, −2.19302643562404637607102922316, −1.81780882985048552637600785011, −1.50764904966784089992766070636, −1.50566936130216917692412130321, −0.880397737145067512000591798564, −0.59419872341747865676429970734, 0.59419872341747865676429970734, 0.880397737145067512000591798564, 1.50566936130216917692412130321, 1.50764904966784089992766070636, 1.81780882985048552637600785011, 2.19302643562404637607102922316, 2.41618162202106074767020392860, 2.54323362755843291156891669940, 3.52081820727396470738447883201, 3.54479439617205875246132175120, 3.62073683677735157437505673779, 3.81165193560217516876460988218, 4.30111191556061734465061944626, 4.42710052881032456329789226494, 4.50343880762503994557043533552, 4.78830838677638834278201643373, 4.92010890859246411725800940399, 5.00487188715650907674742067289, 5.24759176283247482483942251059, 5.41147387494631502569469772070, 5.50004580793099174433454266777, 5.98130236904461519546577517603, 6.07971831909751252502381221942, 6.17404689567621861619154803652, 6.53205321172626881922714777426

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