Properties

Label 8-2352e4-1.1-c3e4-0-2
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $3.70863\times 10^{8}$
Root an. cond. $11.7801$
Motivic weight $3$
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  + 12·3-s − 4·5-s + 90·9-s − 14·11-s − 22·13-s − 48·15-s − 96·17-s − 26·19-s − 96·23-s − 187·25-s + 540·27-s − 76·29-s + 238·31-s − 168·33-s + 562·37-s − 264·39-s − 428·41-s + 258·43-s − 360·45-s − 80·47-s − 1.15e3·51-s + 56·55-s − 312·57-s + 262·59-s + 276·61-s + 88·65-s − 150·67-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.357·5-s + 10/3·9-s − 0.383·11-s − 0.469·13-s − 0.826·15-s − 1.36·17-s − 0.313·19-s − 0.870·23-s − 1.49·25-s + 3.84·27-s − 0.486·29-s + 1.37·31-s − 0.886·33-s + 2.49·37-s − 1.08·39-s − 1.63·41-s + 0.914·43-s − 1.19·45-s − 0.248·47-s − 3.16·51-s + 0.137·55-s − 0.725·57-s + 0.578·59-s + 0.579·61-s + 0.167·65-s − 0.273·67-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.70863\times 10^{8}\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(3.429522935\)
\(L(\frac12)\) \(\approx\) \(3.429522935\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
7 \( 1 \)
good5$C_2 \wr S_4$ \( 1 + 4 T + 203 T^{2} + 1244 T^{3} + 19472 T^{4} + 1244 p^{3} T^{5} + 203 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 14 T - 71 T^{2} - 12014 T^{3} + 2033720 T^{4} - 12014 p^{3} T^{5} - 71 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 22 T + 105 p T^{2} + 60998 T^{3} + 10139200 T^{4} + 60998 p^{3} T^{5} + 105 p^{7} T^{6} + 22 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 96 T + 5348 T^{2} - 301088 T^{3} - 39928698 T^{4} - 301088 p^{3} T^{5} + 5348 p^{6} T^{6} + 96 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 26 T + 13201 T^{2} + 793654 T^{3} + 92486852 T^{4} + 793654 p^{3} T^{5} + 13201 p^{6} T^{6} + 26 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 96 T + 34364 T^{2} + 3625760 T^{3} + 540043494 T^{4} + 3625760 p^{3} T^{5} + 34364 p^{6} T^{6} + 96 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 76 T + 45263 T^{2} + 1864308 T^{3} + 1017389172 T^{4} + 1864308 p^{3} T^{5} + 45263 p^{6} T^{6} + 76 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 238 T + 40396 T^{2} - 4512200 T^{3} + 466839125 T^{4} - 4512200 p^{3} T^{5} + 40396 p^{6} T^{6} - 238 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 562 T + 218917 T^{2} - 51298938 T^{3} + 12421701936 T^{4} - 51298938 p^{3} T^{5} + 218917 p^{6} T^{6} - 562 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 428 T + 234552 T^{2} + 51690756 T^{3} + 18963959182 T^{4} + 51690756 p^{3} T^{5} + 234552 p^{6} T^{6} + 428 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 6 p T + 138485 T^{2} - 14865486 T^{3} + 6723608208 T^{4} - 14865486 p^{3} T^{5} + 138485 p^{6} T^{6} - 6 p^{10} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 80 T + 79044 T^{2} + 6436176 T^{3} + 18721328134 T^{4} + 6436176 p^{3} T^{5} + 79044 p^{6} T^{6} + 80 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 8683 p T^{2} + 3013660 T^{3} + 92692295988 T^{4} + 3013660 p^{3} T^{5} + 8683 p^{7} T^{6} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 262 T + 544965 T^{2} - 206100090 T^{3} + 137845655068 T^{4} - 206100090 p^{3} T^{5} + 544965 p^{6} T^{6} - 262 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 276 T + 472228 T^{2} - 230097500 T^{3} + 121196455014 T^{4} - 230097500 p^{3} T^{5} + 472228 p^{6} T^{6} - 276 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 150 T + 612301 T^{2} + 160051070 T^{3} + 213928110456 T^{4} + 160051070 p^{3} T^{5} + 612301 p^{6} T^{6} + 150 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 848 T + 1353124 T^{2} - 864361552 T^{3} + 714336102950 T^{4} - 864361552 p^{3} T^{5} + 1353124 p^{6} T^{6} - 848 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 218 T + 1524553 T^{2} - 248776730 T^{3} + 12106383700 p T^{4} - 248776730 p^{3} T^{5} + 1524553 p^{6} T^{6} - 218 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1762 T + 2715892 T^{2} + 2479590568 T^{3} + 2098857820333 T^{4} + 2479590568 p^{3} T^{5} + 2715892 p^{6} T^{6} + 1762 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 3450 T + 6524297 T^{2} - 8058006346 T^{3} + 7159809209520 T^{4} - 8058006346 p^{3} T^{5} + 6524297 p^{6} T^{6} - 3450 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 344 T + 1670240 T^{2} - 134058408 T^{3} + 1315175698974 T^{4} - 134058408 p^{3} T^{5} + 1670240 p^{6} T^{6} - 344 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 622 T + 849993 T^{2} + 199355338 T^{3} - 35005181276 T^{4} + 199355338 p^{3} T^{5} + 849993 p^{6} T^{6} - 622 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.10232230444694308002228141400, −5.70802579721869781049949036092, −5.62744685557393856436974677125, −5.47753945225793074171702242928, −5.06068144292289203051744157912, −4.70069378046051312406347341884, −4.60678686698642730106025423198, −4.49377037612110718924872850548, −4.38766330391293568108166617855, −3.89600502488203313208177934209, −3.81642616167790977906240562002, −3.69123381289983874948855385638, −3.54384156520016549761878239828, −2.95854391324749404882833279229, −2.89971402423092069737912219654, −2.73299720152024619801286306303, −2.57522119540594118702356112993, −2.02204257154629439335005955254, −1.91038626894690978397886560573, −1.90780736966263454895888517634, −1.83508609241373917668012029965, −0.963753306978551847029839855192, −0.73953391487158070163236334724, −0.73130621586554670798405678225, −0.12288492316440839237890815973, 0.12288492316440839237890815973, 0.73130621586554670798405678225, 0.73953391487158070163236334724, 0.963753306978551847029839855192, 1.83508609241373917668012029965, 1.90780736966263454895888517634, 1.91038626894690978397886560573, 2.02204257154629439335005955254, 2.57522119540594118702356112993, 2.73299720152024619801286306303, 2.89971402423092069737912219654, 2.95854391324749404882833279229, 3.54384156520016549761878239828, 3.69123381289983874948855385638, 3.81642616167790977906240562002, 3.89600502488203313208177934209, 4.38766330391293568108166617855, 4.49377037612110718924872850548, 4.60678686698642730106025423198, 4.70069378046051312406347341884, 5.06068144292289203051744157912, 5.47753945225793074171702242928, 5.62744685557393856436974677125, 5.70802579721869781049949036092, 6.10232230444694308002228141400

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