Properties

Label 8-363e4-1.1-c5e4-0-1
Degree $8$
Conductor $17363069361$
Sign $1$
Analytic cond. $1.14886\times 10^{7}$
Root an. cond. $7.63015$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·2-s − 36·3-s + 3·4-s + 42·5-s − 324·6-s − 14·7-s − 57·8-s + 810·9-s + 378·10-s − 108·12-s + 886·13-s − 126·14-s − 1.51e3·15-s + 1.05e3·16-s − 570·17-s + 7.29e3·18-s + 1.33e3·19-s + 126·20-s + 504·21-s + 1.26e3·23-s + 2.05e3·24-s − 6.14e3·25-s + 7.97e3·26-s − 1.45e4·27-s − 42·28-s + 1.02e4·29-s − 1.36e4·30-s + ⋯
L(s)  = 1  + 1.59·2-s − 2.30·3-s + 3/32·4-s + 0.751·5-s − 3.67·6-s − 0.107·7-s − 0.314·8-s + 10/3·9-s + 1.19·10-s − 0.216·12-s + 1.45·13-s − 0.171·14-s − 1.73·15-s + 1.02·16-s − 0.478·17-s + 5.30·18-s + 0.850·19-s + 0.0704·20-s + 0.249·21-s + 0.496·23-s + 0.727·24-s − 1.96·25-s + 2.31·26-s − 3.84·27-s − 0.0101·28-s + 2.25·29-s − 2.76·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.14886\times 10^{7}\)
Root analytic conductor: \(7.63015\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(8.481298406\)
\(L(\frac12)\) \(\approx\) \(8.481298406\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{4} \)
11 \( 1 \)
good2$C_2 \wr S_4$ \( 1 - 9 T + 39 p T^{2} - 309 p T^{3} + 941 p^{2} T^{4} - 309 p^{6} T^{5} + 39 p^{11} T^{6} - 9 p^{15} T^{7} + p^{20} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 42 T + 1581 p T^{2} - 298638 T^{3} + 35774024 T^{4} - 298638 p^{5} T^{5} + 1581 p^{11} T^{6} - 42 p^{15} T^{7} + p^{20} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 2 p T + 25261 T^{2} - 2055082 T^{3} + 344115460 T^{4} - 2055082 p^{5} T^{5} + 25261 p^{10} T^{6} + 2 p^{16} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 886 T + 809461 T^{2} - 496080238 T^{3} + 405938439172 T^{4} - 496080238 p^{5} T^{5} + 809461 p^{10} T^{6} - 886 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 570 T + 2270085 T^{2} - 1804796634 T^{3} + 1078768320248 T^{4} - 1804796634 p^{5} T^{5} + 2270085 p^{10} T^{6} + 570 p^{15} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 1338 T + 7542517 T^{2} - 10474242858 T^{3} + 25064736574956 T^{4} - 10474242858 p^{5} T^{5} + 7542517 p^{10} T^{6} - 1338 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 1260 T + 7754880 T^{2} - 12979708092 T^{3} + 88859643423326 T^{4} - 12979708092 p^{5} T^{5} + 7754880 p^{10} T^{6} - 1260 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 10230 T + 93323813 T^{2} - 475634451006 T^{3} + 2588636637314532 T^{4} - 475634451006 p^{5} T^{5} + 93323813 p^{10} T^{6} - 10230 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 8042 T + 94978261 T^{2} + 417964095266 T^{3} + 3289572375041044 T^{4} + 417964095266 p^{5} T^{5} + 94978261 p^{10} T^{6} + 8042 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 18936 T + 376349398 T^{2} - 3936210595416 T^{3} + 41815812920398815 T^{4} - 3936210595416 p^{5} T^{5} + 376349398 p^{10} T^{6} - 18936 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 3006 T + 192374697 T^{2} + 651736053318 T^{3} + 33401079864830084 T^{4} + 651736053318 p^{5} T^{5} + 192374697 p^{10} T^{6} + 3006 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 21504 T + 294848956 T^{2} + 1575316024704 T^{3} + 10479524307735414 T^{4} + 1575316024704 p^{5} T^{5} + 294848956 p^{10} T^{6} + 21504 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 5916 T + 433873104 T^{2} + 302419745076 T^{3} + 94994293024440158 T^{4} + 302419745076 p^{5} T^{5} + 433873104 p^{10} T^{6} - 5916 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 48414 T + 1901445533 T^{2} - 54658035953766 T^{3} + 1202239744227438084 T^{4} - 54658035953766 p^{5} T^{5} + 1901445533 p^{10} T^{6} - 48414 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 30276 T + 2135304432 T^{2} - 44131848488100 T^{3} + 2027676021116746670 T^{4} - 44131848488100 p^{5} T^{5} + 2135304432 p^{10} T^{6} - 30276 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 106242 T + 6532906537 T^{2} - 280841840520642 T^{3} + 9186780257728975740 T^{4} - 280841840520642 p^{5} T^{5} + 6532906537 p^{10} T^{6} - 106242 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 57538 T + 4416412861 T^{2} + 157787648605138 T^{3} + 7497513505176486892 T^{4} + 157787648605138 p^{5} T^{5} + 4416412861 p^{10} T^{6} + 57538 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 45720 T + 4522588316 T^{2} - 205135961766264 T^{3} + 10871340808413470886 T^{4} - 205135961766264 p^{5} T^{5} + 4522588316 p^{10} T^{6} - 45720 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 11426 T + 7374900745 T^{2} + 74076180419162 T^{3} + 22044509882838983620 T^{4} + 74076180419162 p^{5} T^{5} + 7374900745 p^{10} T^{6} + 11426 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 68338 T + 8930204917 T^{2} + 465489883497874 T^{3} + 38131158729353014276 T^{4} + 465489883497874 p^{5} T^{5} + 8930204917 p^{10} T^{6} + 68338 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 146748 T + 14767913360 T^{2} - 834109700912028 T^{3} + 53639101259952424782 T^{4} - 834109700912028 p^{5} T^{5} + 14767913360 p^{10} T^{6} - 146748 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 89106 T + 23226988445 T^{2} + 1474906789358334 T^{3} + \)\(19\!\cdots\!44\)\( T^{4} + 1474906789358334 p^{5} T^{5} + 23226988445 p^{10} T^{6} + 89106 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 386120 T + 77496909178 T^{2} - 10297690878548432 T^{3} + \)\(10\!\cdots\!11\)\( T^{4} - 10297690878548432 p^{5} T^{5} + 77496909178 p^{10} T^{6} - 386120 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.19317203996547512630647955034, −6.90669943975363669787374490068, −6.78253420963731683610399331938, −6.46466436034316409313634973305, −6.38677944690884127083125884536, −5.76036769558754185011842202954, −5.71022681483976054599910521085, −5.62724298022419297687828289688, −5.58230073861143375971600209649, −4.89935894520753533258873768669, −4.80924765150455302547940031310, −4.74263180784138570155526182701, −4.48618552155714299169870263654, −3.96399415714268806939639852084, −3.71236823384895403195905263623, −3.67053890194210438584820417021, −3.50890189281722898819827423270, −2.54861098349105589372600114707, −2.51405344708946524446134706989, −1.94559407280401730385351462029, −1.49332229056283671617434408366, −1.37038372239063851046110868507, −0.818501750030732924738244287346, −0.56835558783561959844875224983, −0.44349478808048304634694234204, 0.44349478808048304634694234204, 0.56835558783561959844875224983, 0.818501750030732924738244287346, 1.37038372239063851046110868507, 1.49332229056283671617434408366, 1.94559407280401730385351462029, 2.51405344708946524446134706989, 2.54861098349105589372600114707, 3.50890189281722898819827423270, 3.67053890194210438584820417021, 3.71236823384895403195905263623, 3.96399415714268806939639852084, 4.48618552155714299169870263654, 4.74263180784138570155526182701, 4.80924765150455302547940031310, 4.89935894520753533258873768669, 5.58230073861143375971600209649, 5.62724298022419297687828289688, 5.71022681483976054599910521085, 5.76036769558754185011842202954, 6.38677944690884127083125884536, 6.46466436034316409313634973305, 6.78253420963731683610399331938, 6.90669943975363669787374490068, 7.19317203996547512630647955034

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