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Properties

Label 8-1045e4-1.1-c0e4-0-0
Degree $8$
Conductor $1.193\times 10^{12}$
Sign $1$
Analytic cond. $0.0739764$
Root an. cond. $0.722165$
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  − 2·2-s − 2·3-s + 4-s − 5-s + 4·6-s + 9-s + 2·10-s − 11-s − 2·12-s − 2·13-s + 2·15-s − 2·18-s − 19-s − 20-s + 2·22-s + 4·26-s − 4·30-s + 2·32-s + 2·33-s + 36-s − 2·37-s + 2·38-s + 4·39-s − 44-s − 45-s − 49-s − 2·52-s + ⋯
L(s)  = 1  − 2·2-s − 2·3-s + 4-s − 5-s + 4·6-s + 9-s + 2·10-s − 11-s − 2·12-s − 2·13-s + 2·15-s − 2·18-s − 19-s − 20-s + 2·22-s + 4·26-s − 4·30-s + 2·32-s + 2·33-s + 36-s − 2·37-s + 2·38-s + 4·39-s − 44-s − 45-s − 49-s − 2·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 11^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(0.0739764\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 11^{4} \cdot 19^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005520592410\)
\(L(\frac12)\) \(\approx\) \(0.005520592410\)
\(L(1)\) 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)$
bad5$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
19$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good2$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
3$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
7$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17$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_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
31$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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$$\times$$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$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
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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

−7.22958987224353223406111757089, −7.18013880145692810304374731024, −7.15355582849844669644080438137, −6.54233629253998208464116970160, −6.47137403209915347191360585446, −6.27380657632925068812619422060, −6.03516447731750310650178453806, −5.88313498391416528075764014017, −5.35214888966875717031487586100, −5.27707086646491639640642140796, −5.04628685002785603161654090759, −5.00526866480211695744943522114, −4.76538053045065270177955006570, −4.33395864518159594837999363340, −4.08275180719986615025978957516, −3.89082791983339165068547208944, −3.60254345950270701183754070118, −2.97536579047653453007591549812, −2.86948013701621776684247653956, −2.65286351502114337872849611534, −2.26130537906733152861242861994, −1.74387819764198111530161953332, −1.47472913302068063149451516290, −0.57402276687969169064111777264, −0.17732137181966874075500306757, 0.17732137181966874075500306757, 0.57402276687969169064111777264, 1.47472913302068063149451516290, 1.74387819764198111530161953332, 2.26130537906733152861242861994, 2.65286351502114337872849611534, 2.86948013701621776684247653956, 2.97536579047653453007591549812, 3.60254345950270701183754070118, 3.89082791983339165068547208944, 4.08275180719986615025978957516, 4.33395864518159594837999363340, 4.76538053045065270177955006570, 5.00526866480211695744943522114, 5.04628685002785603161654090759, 5.27707086646491639640642140796, 5.35214888966875717031487586100, 5.88313498391416528075764014017, 6.03516447731750310650178453806, 6.27380657632925068812619422060, 6.47137403209915347191360585446, 6.54233629253998208464116970160, 7.15355582849844669644080438137, 7.18013880145692810304374731024, 7.22958987224353223406111757089

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