Properties

Label 8-1232e4-1.1-c0e4-0-0
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $0.142912$
Root an. cond. $0.784122$
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  + 7-s − 9-s + 11-s + 2·23-s − 25-s − 2·29-s + 3·37-s + 2·43-s + 3·53-s − 63-s + 2·67-s − 3·71-s + 77-s − 3·79-s − 99-s − 3·107-s − 2·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·161-s + 163-s + ⋯
L(s)  = 1  + 7-s − 9-s + 11-s + 2·23-s − 25-s − 2·29-s + 3·37-s + 2·43-s + 3·53-s − 63-s + 2·67-s − 3·71-s + 77-s − 3·79-s − 99-s − 3·107-s − 2·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·161-s + 163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(0.142912\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.196055532\)
\(L(\frac12)\) \(\approx\) \(1.196055532\)
\(L(1)\) 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 \( 1 \)
7$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
11$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
good3$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
5$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$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
29$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
41$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
67$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
71$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
73$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
79$C_1$$\times$$C_4$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
83$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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

−7.26621156211003848682512437511, −6.91499702183935283184843071122, −6.77063157495418659267574525537, −6.48464846713364718473076730065, −6.30748084129389519889889979420, −5.83341835388384917405537368685, −5.76460682695326382548566634948, −5.67793691137486277400329464511, −5.55278788370498961833474498219, −5.19020497070818483116764411460, −4.84392893199080262301725766528, −4.77256884090935572531406690547, −4.34157952251865343471472497318, −4.10400001206323126466211764423, −3.95118965721499041563046368770, −3.72020678993692712824848195598, −3.70157548404495699206300376854, −2.86058795416256148059944636157, −2.72099764306374197918307625911, −2.59591746557874268477463706604, −2.58950472534529719761011927287, −1.83203263244261924111559821474, −1.53854040227730921481415342678, −1.26563810431600125708071843424, −0.840032564715966728345163630414, 0.840032564715966728345163630414, 1.26563810431600125708071843424, 1.53854040227730921481415342678, 1.83203263244261924111559821474, 2.58950472534529719761011927287, 2.59591746557874268477463706604, 2.72099764306374197918307625911, 2.86058795416256148059944636157, 3.70157548404495699206300376854, 3.72020678993692712824848195598, 3.95118965721499041563046368770, 4.10400001206323126466211764423, 4.34157952251865343471472497318, 4.77256884090935572531406690547, 4.84392893199080262301725766528, 5.19020497070818483116764411460, 5.55278788370498961833474498219, 5.67793691137486277400329464511, 5.76460682695326382548566634948, 5.83341835388384917405537368685, 6.30748084129389519889889979420, 6.48464846713364718473076730065, 6.77063157495418659267574525537, 6.91499702183935283184843071122, 7.26621156211003848682512437511

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