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Properties

Label 8-363e4-1.1-c0e4-0-0
Degree $8$
Conductor $17363069361$
Sign $1$
Analytic cond. $0.00107709$
Root an. cond. $0.425629$
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  + 3-s − 4-s − 12-s − 25-s + 2·31-s + 2·37-s + 49-s − 8·67-s − 75-s + 2·93-s − 2·97-s + 100-s − 2·103-s + 2·111-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 147-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯
L(s)  = 1  + 3-s − 4-s − 12-s − 25-s + 2·31-s + 2·37-s + 49-s − 8·67-s − 75-s + 2·93-s − 2·97-s + 100-s − 2·103-s + 2·111-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 147-s − 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(0.00107709\)
Root analytic conductor: \(0.425629\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4537589557\)
\(L(\frac12)\) \(\approx\) \(0.4537589557\)
\(L(1)\) 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)$
bad3$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
11 \( 1 \)
good2$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$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
7$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
13$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
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$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{4} \)
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$$\times$$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$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
67$C_1$ \( ( 1 + T )^{8} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
79$C_4\times C_2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
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

−8.568738621602930742608859516826, −8.344628753986667870938261585630, −8.068266425726644965230629538474, −7.925887670773080368600918124268, −7.55258136841995786417402856092, −7.47670976419178148087923270472, −7.12695092971005917993398219619, −6.80864664172881355213017516475, −6.56650594185616066295002626321, −6.08849349435628478958587913532, −5.81687527139242650830957661593, −5.80426373096249729933395900283, −5.71398631229508462563979728516, −4.78938295323439278206713657386, −4.75801903048621441368532879668, −4.51339613770582073786590703460, −4.45532730895249826227240071082, −3.81831537282911739701181412907, −3.81368417415172572109019823283, −3.23112915873196011125024679814, −2.73899493222222187686912804421, −2.69886119450899712520019757737, −2.51911550497708430071669515855, −1.54848778935769911800562078805, −1.34966010933792512130866683765, 1.34966010933792512130866683765, 1.54848778935769911800562078805, 2.51911550497708430071669515855, 2.69886119450899712520019757737, 2.73899493222222187686912804421, 3.23112915873196011125024679814, 3.81368417415172572109019823283, 3.81831537282911739701181412907, 4.45532730895249826227240071082, 4.51339613770582073786590703460, 4.75801903048621441368532879668, 4.78938295323439278206713657386, 5.71398631229508462563979728516, 5.80426373096249729933395900283, 5.81687527139242650830957661593, 6.08849349435628478958587913532, 6.56650594185616066295002626321, 6.80864664172881355213017516475, 7.12695092971005917993398219619, 7.47670976419178148087923270472, 7.55258136841995786417402856092, 7.925887670773080368600918124268, 8.068266425726644965230629538474, 8.344628753986667870938261585630, 8.568738621602930742608859516826

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