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Properties

Label 8-77e4-1.1-c0e4-0-0
Degree $8$
Conductor $35153041$
Sign $1$
Analytic cond. $2.18067\times 10^{-6}$
Root an. cond. $0.196030$
Motivic weight $0$
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  − 2·2-s + 4-s − 7-s − 9-s − 11-s + 2·14-s + 2·18-s + 2·22-s − 2·23-s − 25-s − 28-s − 2·29-s + 2·32-s − 36-s + 3·37-s − 2·43-s − 44-s + 4·46-s + 2·50-s + 3·53-s + 4·58-s + 63-s − 4·64-s − 2·67-s + 3·71-s − 6·74-s + 77-s + ⋯
L(s)  = 1  − 2·2-s + 4-s − 7-s − 9-s − 11-s + 2·14-s + 2·18-s + 2·22-s − 2·23-s − 25-s − 28-s − 2·29-s + 2·32-s − 36-s + 3·37-s − 2·43-s − 44-s + 4·46-s + 2·50-s + 3·53-s + 4·58-s + 63-s − 4·64-s − 2·67-s + 3·71-s − 6·74-s + 77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(35153041\)    =    \(7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2.18067\times 10^{-6}\)
Root analytic conductor: \(0.196030\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 35153041,\ (\ :0, 0, 0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02364688051\)
\(L(\frac12)\) \(\approx\) \(0.02364688051\)
\(L(1)\) 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)$
bad7$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good2$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
3$C_4$$\times$$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

−11.14297921968316790348669077014, −10.45975473868778616407928874431, −10.14236648571518692718561215438, −10.12551310524947616653456690813, −9.911879241447849028762294713107, −9.414322799362812733857412722582, −9.212076943517285374974722917022, −9.096411228983060337014424351550, −8.935139995367423919312740922857, −8.145878264529248582217719935070, −8.036248377990937902874141406961, −7.997295040882863506221465232122, −7.86864847338819353586449231672, −7.03454853540565329256094827597, −6.88058601121399683445656354047, −6.14991031457579889100996880776, −5.99237450344020077548331918694, −5.98104918423109067487880098106, −5.16992084082465797227471799327, −5.02305203641943809312723389715, −4.11400874670323536123765252069, −3.89149719521873110298670825170, −3.35405452155751025316205560791, −2.48329704472032446623488429724, −2.35278045266079068339323690453, 2.35278045266079068339323690453, 2.48329704472032446623488429724, 3.35405452155751025316205560791, 3.89149719521873110298670825170, 4.11400874670323536123765252069, 5.02305203641943809312723389715, 5.16992084082465797227471799327, 5.98104918423109067487880098106, 5.99237450344020077548331918694, 6.14991031457579889100996880776, 6.88058601121399683445656354047, 7.03454853540565329256094827597, 7.86864847338819353586449231672, 7.997295040882863506221465232122, 8.036248377990937902874141406961, 8.145878264529248582217719935070, 8.935139995367423919312740922857, 9.096411228983060337014424351550, 9.212076943517285374974722917022, 9.414322799362812733857412722582, 9.911879241447849028762294713107, 10.12551310524947616653456690813, 10.14236648571518692718561215438, 10.45975473868778616407928874431, 11.14297921968316790348669077014

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