Properties

Label 8-2352e4-1.1-c2e4-0-0
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $1.68690\times 10^{7}$
Root an. cond. $8.00545$
Motivic weight $2$
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  − 2·5-s − 6·9-s + 26·13-s − 40·17-s − 69·25-s − 2·29-s + 38·37-s − 4·41-s + 12·45-s − 338·53-s − 192·61-s − 52·65-s + 382·73-s + 27·81-s + 80·85-s − 244·89-s + 50·97-s − 208·101-s − 490·109-s − 500·113-s − 156·117-s + 125·121-s + 162·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2/5·5-s − 2/3·9-s + 2·13-s − 2.35·17-s − 2.75·25-s − 0.0689·29-s + 1.02·37-s − 0.0975·41-s + 4/15·45-s − 6.37·53-s − 3.14·61-s − 4/5·65-s + 5.23·73-s + 1/3·81-s + 0.941·85-s − 2.74·89-s + 0.515·97-s − 2.05·101-s − 4.49·109-s − 4.42·113-s − 4/3·117-s + 1.03·121-s + 1.29·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.68690\times 10^{7}\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.003888881049\)
\(L(\frac12)\) \(\approx\) \(0.003888881049\)
\(L(2)\) 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)$
bad2 \( 1 \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( ( 1 + T + 36 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 125 T^{2} + 4332 T^{4} - 125 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 - p T + 366 T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 20 T + 450 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 175 T^{2} + 246624 T^{4} + 175 p^{4} T^{6} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 1940 T^{2} + 1496934 T^{4} - 1940 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 + T - 42 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 1126 T^{2} + 317211 T^{4} - 1126 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 19 T + 2130 T^{2} - 19 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 2 T + 1938 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 3557 T^{2} + 9145308 T^{4} - 3557 p^{4} T^{6} + p^{8} T^{8} \)
47$C_2^2$ \( ( 1 - 3830 T^{2} + p^{4} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 169 T + 234 p T^{2} + 169 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 11057 T^{2} + 54465456 T^{4} - 11057 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 96 T + 9518 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 17173 T^{2} + 114019836 T^{4} - 17173 p^{4} T^{6} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 4748 T^{2} + 53176038 T^{4} - 4748 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 191 T + 19764 T^{2} - 191 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 17182 T^{2} + 137226531 T^{4} - 17182 p^{4} T^{6} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 1997 T^{2} - 7277460 T^{4} - 1997 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 + 122 T + 19506 T^{2} + 122 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 25 T + 5280 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−6.26223578370202680334539984875, −6.02999106809509438965577673681, −5.91684488472602107696242703640, −5.67753181669144465426118653671, −5.31256933929846170257888713187, −5.05376771336871339474876722617, −5.03249454936311877596755551425, −4.66730585504595869932009494517, −4.38561654736545689488406627339, −4.25985956180035149288596039951, −4.15766388350662701269381139687, −3.70826633056065748895676598387, −3.65467920325767969404902855203, −3.38344517367198712281954991819, −3.31030628734734508677647587005, −2.83092708694955386537606054568, −2.45445249340438594214879875518, −2.44811261441761513921389402851, −2.23338910436891308748945266760, −1.69906715434846442329005155019, −1.36819334565457464376082375113, −1.36152672137530960174749568608, −1.13907772610078694357918020074, −0.07336589365149625320631155984, −0.05123440773356589925177204041, 0.05123440773356589925177204041, 0.07336589365149625320631155984, 1.13907772610078694357918020074, 1.36152672137530960174749568608, 1.36819334565457464376082375113, 1.69906715434846442329005155019, 2.23338910436891308748945266760, 2.44811261441761513921389402851, 2.45445249340438594214879875518, 2.83092708694955386537606054568, 3.31030628734734508677647587005, 3.38344517367198712281954991819, 3.65467920325767969404902855203, 3.70826633056065748895676598387, 4.15766388350662701269381139687, 4.25985956180035149288596039951, 4.38561654736545689488406627339, 4.66730585504595869932009494517, 5.03249454936311877596755551425, 5.05376771336871339474876722617, 5.31256933929846170257888713187, 5.67753181669144465426118653671, 5.91684488472602107696242703640, 6.02999106809509438965577673681, 6.26223578370202680334539984875

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