Properties

Label 8-48e4-1.1-c18e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $9.44603\times 10^{7}$
Root an. cond. $9.92901$
Motivic weight $18$
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  − 1.58e4·3-s + 9.57e7·7-s − 3.16e8·9-s − 5.42e9·13-s − 1.91e11·19-s − 1.52e12·21-s + 1.33e13·25-s + 7.90e12·27-s − 3.57e13·31-s − 4.75e14·37-s + 8.61e13·39-s − 1.59e15·43-s − 3.85e14·49-s + 3.03e15·57-s − 2.20e15·61-s − 3.03e16·63-s + 4.13e16·67-s − 4.53e16·73-s − 2.11e17·75-s + 1.77e17·79-s + 2.12e15·81-s − 5.19e17·91-s + 5.67e17·93-s − 3.49e18·97-s + 1.25e18·103-s + 5.25e17·109-s + 7.54e18·111-s + ⋯
L(s)  = 1  − 0.806·3-s + 2.37·7-s − 0.817·9-s − 0.511·13-s − 0.593·19-s − 1.91·21-s + 3.49·25-s + 1.03·27-s − 1.35·31-s − 3.65·37-s + 0.412·39-s − 3.18·43-s − 0.236·49-s + 0.478·57-s − 0.188·61-s − 1.93·63-s + 1.51·67-s − 0.770·73-s − 2.81·75-s + 1.47·79-s + 0.0141·81-s − 1.21·91-s + 1.08·93-s − 4.59·97-s + 0.958·103-s + 0.241·109-s + 2.94·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+9)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(9.44603\times 10^{7}\)
Root analytic conductor: \(9.92901\)
Motivic weight: \(18\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :9, 9, 9, 9),\ 1)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.7238693100\)
\(L(\frac12)\) \(\approx\) \(0.7238693100\)
\(L(10)\) 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 \( 1 \)
3$D_{4}$ \( 1 + 196 p^{4} T + 86678 p^{8} T^{2} + 196 p^{22} T^{3} + p^{36} T^{4} \)
good5$C_2^2 \wr C_2$ \( 1 - 2666638851668 p T^{2} + \)\(23\!\cdots\!82\)\( p^{5} T^{4} - 2666638851668 p^{37} T^{6} + p^{72} T^{8} \)
7$D_{4}$ \( ( 1 - 6838868 p T + 10583755945386 p^{3} T^{2} - 6838868 p^{19} T^{3} + p^{36} T^{4} )^{2} \)
11$C_2^2 \wr C_2$ \( 1 - 8040762618362030884 T^{2} + \)\(59\!\cdots\!66\)\( p^{2} T^{4} - 8040762618362030884 p^{36} T^{6} + p^{72} T^{8} \)
13$D_{4}$ \( ( 1 + 208700828 p T + 285201747815178822 p^{2} T^{2} + 208700828 p^{19} T^{3} + p^{36} T^{4} )^{2} \)
17$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!20\)\( p^{2} T^{2} + \)\(71\!\cdots\!58\)\( p^{4} T^{4} - \)\(12\!\cdots\!20\)\( p^{38} T^{6} + p^{72} T^{8} \)
19$D_{4}$ \( ( 1 + 95708324740 T + \)\(12\!\cdots\!18\)\( T^{2} + 95708324740 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
23$C_2^2 \wr C_2$ \( 1 - \)\(89\!\cdots\!60\)\( T^{2} + \)\(38\!\cdots\!18\)\( T^{4} - \)\(89\!\cdots\!60\)\( p^{36} T^{6} + p^{72} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 - \)\(53\!\cdots\!84\)\( T^{2} + \)\(16\!\cdots\!06\)\( T^{4} - \)\(53\!\cdots\!84\)\( p^{36} T^{6} + p^{72} T^{8} \)
31$D_{4}$ \( ( 1 + 17864207542804 T + \)\(10\!\cdots\!86\)\( T^{2} + 17864207542804 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 237649916751116 T + \)\(47\!\cdots\!98\)\( T^{2} + 237649916751116 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 - \)\(21\!\cdots\!44\)\( T^{2} + \)\(34\!\cdots\!66\)\( T^{4} - \)\(21\!\cdots\!44\)\( p^{36} T^{6} + p^{72} T^{8} \)
43$D_{4}$ \( ( 1 + 799803662287396 T + \)\(47\!\cdots\!98\)\( T^{2} + 799803662287396 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
47$C_2^2 \wr C_2$ \( 1 - \)\(33\!\cdots\!40\)\( T^{2} + \)\(59\!\cdots\!18\)\( T^{4} - \)\(33\!\cdots\!40\)\( p^{36} T^{6} + p^{72} T^{8} \)
53$C_2^2 \wr C_2$ \( 1 - \)\(13\!\cdots\!40\)\( T^{2} + \)\(66\!\cdots\!18\)\( T^{4} - \)\(13\!\cdots\!40\)\( p^{36} T^{6} + p^{72} T^{8} \)
59$C_2^2 \wr C_2$ \( 1 - \)\(28\!\cdots\!44\)\( T^{2} + \)\(32\!\cdots\!66\)\( T^{4} - \)\(28\!\cdots\!44\)\( p^{36} T^{6} + p^{72} T^{8} \)
61$D_{4}$ \( ( 1 + 1103634372941996 T + \)\(15\!\cdots\!66\)\( T^{2} + 1103634372941996 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 20665768039464956 T + \)\(15\!\cdots\!98\)\( T^{2} - 20665768039464956 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
71$C_2^2 \wr C_2$ \( 1 - \)\(35\!\cdots\!84\)\( T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - \)\(35\!\cdots\!84\)\( p^{36} T^{6} + p^{72} T^{8} \)
73$D_{4}$ \( ( 1 + 22694977381228124 T + \)\(62\!\cdots\!78\)\( T^{2} + 22694977381228124 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 1122565721708180 p T + \)\(29\!\cdots\!58\)\( T^{2} - 1122565721708180 p^{19} T^{3} + p^{36} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 - \)\(12\!\cdots\!40\)\( T^{2} + \)\(64\!\cdots\!98\)\( T^{4} - \)\(12\!\cdots\!40\)\( p^{36} T^{6} + p^{72} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - \)\(22\!\cdots\!84\)\( T^{2} + \)\(25\!\cdots\!86\)\( T^{4} - \)\(22\!\cdots\!84\)\( p^{36} T^{6} + p^{72} T^{8} \)
97$D_{4}$ \( ( 1 + 1745782267257380156 T + \)\(18\!\cdots\!98\)\( T^{2} + 1745782267257380156 p^{18} T^{3} + p^{36} T^{4} )^{2} \)
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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

−8.365488827986960827708713940639, −7.53563596976364290876005512821, −7.44977173794639795529610419571, −7.08839923078443221346251730535, −6.50533159386623440404702370134, −6.49911200156412392365008120153, −6.47508732335417893246604704934, −5.36454671983690044792953535296, −5.32383151636541091569399921476, −5.18316275634013837387994096949, −5.15044717802927144992111238919, −4.69433362349414773277879108868, −4.43704685895901354723939622276, −3.99643409243199876526046299693, −3.30266302234712167870813276781, −3.20120055875016684965873252688, −3.15531988696148786712147648688, −2.37537340002313546155608578724, −2.05632963500589160274482712651, −1.88686714609136446133804576631, −1.41286252978331921078221603465, −1.26960945782549159317799853345, −1.04171418008527356139289278084, −0.24932509391087531165908801694, −0.18837114781025652421936738359, 0.18837114781025652421936738359, 0.24932509391087531165908801694, 1.04171418008527356139289278084, 1.26960945782549159317799853345, 1.41286252978331921078221603465, 1.88686714609136446133804576631, 2.05632963500589160274482712651, 2.37537340002313546155608578724, 3.15531988696148786712147648688, 3.20120055875016684965873252688, 3.30266302234712167870813276781, 3.99643409243199876526046299693, 4.43704685895901354723939622276, 4.69433362349414773277879108868, 5.15044717802927144992111238919, 5.18316275634013837387994096949, 5.32383151636541091569399921476, 5.36454671983690044792953535296, 6.47508732335417893246604704934, 6.49911200156412392365008120153, 6.50533159386623440404702370134, 7.08839923078443221346251730535, 7.44977173794639795529610419571, 7.53563596976364290876005512821, 8.365488827986960827708713940639

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