Properties

Label 8-507e4-1.1-c1e4-0-0
Degree $8$
Conductor $66074188401$
Sign $1$
Analytic cond. $268.621$
Root an. cond. $2.01206$
Motivic weight $1$
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  + 2·7-s + 6·9-s − 8·16-s − 16·19-s − 14·31-s − 20·37-s + 2·49-s + 12·63-s − 22·67-s − 34·73-s + 27·81-s + 10·97-s + 38·109-s − 16·112-s + 127-s + 131-s − 32·133-s + 137-s + 139-s − 48·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 96·171-s + 173-s + ⋯
L(s)  = 1  + 0.755·7-s + 2·9-s − 2·16-s − 3.67·19-s − 2.51·31-s − 3.28·37-s + 2/7·49-s + 1.51·63-s − 2.68·67-s − 3.97·73-s + 3·81-s + 1.01·97-s + 3.63·109-s − 1.51·112-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s − 4·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 7.34·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(268.621\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{507} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2814745716\)
\(L(\frac12)\) \(\approx\) \(0.2814745716\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
13 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
5$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
7$C_2$$\times$$C_2^2$ \( ( 1 - T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \)
11$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$$\times$$C_2^2$ \( ( 1 + 8 T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \)
37$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \)
41$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \)
47$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 47 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 11 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \)
71$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
73$C_2$$\times$$C_2^2$ \( ( 1 + 17 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 - 169 T^{2} + p^{2} T^{4} ) \)
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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

−7.62144378737474454015191852309, −7.43921663760680981755768897117, −7.38318813135460189955454378731, −7.37004458763722651040887815706, −6.89921467456461448234756600202, −6.76835648053072734459229328095, −6.49141196469289806162188124450, −6.13600869747433017216839382978, −5.94850236425707780040984914186, −5.91949938385819876397981869923, −5.19014445140734830202037017848, −4.85577079397094602955820392245, −4.80105152107864745881265526075, −4.79241584067534060575735726657, −4.20998871526297960127435630745, −4.08615422428834798132064129461, −3.79268767175375804761478461030, −3.68982284830079010331445129948, −3.10626912948006536621911433562, −2.49641399468138284479520037360, −2.35783233922793297285974502560, −1.70367055843118625946056654726, −1.67722203884899974366815108604, −1.65867033055484333747565087492, −0.15698818360259769906684857039, 0.15698818360259769906684857039, 1.65867033055484333747565087492, 1.67722203884899974366815108604, 1.70367055843118625946056654726, 2.35783233922793297285974502560, 2.49641399468138284479520037360, 3.10626912948006536621911433562, 3.68982284830079010331445129948, 3.79268767175375804761478461030, 4.08615422428834798132064129461, 4.20998871526297960127435630745, 4.79241584067534060575735726657, 4.80105152107864745881265526075, 4.85577079397094602955820392245, 5.19014445140734830202037017848, 5.91949938385819876397981869923, 5.94850236425707780040984914186, 6.13600869747433017216839382978, 6.49141196469289806162188124450, 6.76835648053072734459229328095, 6.89921467456461448234756600202, 7.37004458763722651040887815706, 7.38318813135460189955454378731, 7.43921663760680981755768897117, 7.62144378737474454015191852309

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