Properties

Label 8-363e4-1.1-c7e4-0-0
Degree $8$
Conductor $17363069361$
Sign $1$
Analytic cond. $1.65343\times 10^{8}$
Root an. cond. $10.6487$
Motivic weight $7$
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  − 15·2-s + 108·3-s + 139·4-s + 306·5-s − 1.62e3·6-s − 890·7-s − 1.13e3·8-s + 7.29e3·9-s − 4.59e3·10-s + 1.50e4·12-s + 1.82e3·13-s + 1.33e4·14-s + 3.30e4·15-s + 6.30e3·16-s − 3.28e4·17-s − 1.09e5·18-s + 1.27e4·19-s + 4.25e4·20-s − 9.61e4·21-s + 1.14e5·23-s − 1.23e5·24-s − 7.30e4·25-s − 2.73e4·26-s + 3.93e5·27-s − 1.23e5·28-s + 1.04e5·29-s − 4.95e5·30-s + ⋯
L(s)  = 1  − 1.32·2-s + 2.30·3-s + 1.08·4-s + 1.09·5-s − 3.06·6-s − 0.980·7-s − 0.786·8-s + 10/3·9-s − 1.45·10-s + 2.50·12-s + 0.230·13-s + 1.30·14-s + 2.52·15-s + 0.384·16-s − 1.62·17-s − 4.41·18-s + 0.427·19-s + 1.18·20-s − 2.26·21-s + 1.96·23-s − 1.81·24-s − 0.934·25-s − 0.304·26-s + 3.84·27-s − 1.06·28-s + 0.799·29-s − 3.35·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.65343\times 10^{8}\)
Root analytic conductor: \(10.6487\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(12.45445229\)
\(L(\frac12)\) \(\approx\) \(12.45445229\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 - p^{3} T )^{4} \)
11 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + 15 T + 43 p T^{2} + 43 p^{3} T^{3} + 249 p^{4} T^{4} + 43 p^{10} T^{5} + 43 p^{15} T^{6} + 15 p^{21} T^{7} + p^{28} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 306 T + 33328 p T^{2} - 1192838 p^{2} T^{3} + 120655398 p^{3} T^{4} - 1192838 p^{9} T^{5} + 33328 p^{15} T^{6} - 306 p^{21} T^{7} + p^{28} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 890 T + 2428648 T^{2} + 1513098682 T^{3} + 2713357113806 T^{4} + 1513098682 p^{7} T^{5} + 2428648 p^{14} T^{6} + 890 p^{21} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 1822 T + 143434048 T^{2} - 57178087826 p T^{3} + 9963947917232846 T^{4} - 57178087826 p^{8} T^{5} + 143434048 p^{14} T^{6} - 1822 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 32856 T + 1617694640 T^{2} + 2092957427944 p T^{3} + 1006917500477444958 T^{4} + 2092957427944 p^{8} T^{5} + 1617694640 p^{14} T^{6} + 32856 p^{21} T^{7} + p^{28} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 12784 T + 1968859072 T^{2} - 47795746564128 T^{3} + 103428707615650746 p T^{4} - 47795746564128 p^{7} T^{5} + 1968859072 p^{14} T^{6} - 12784 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 114858 T + 9605711756 T^{2} - 598055876188978 T^{3} + 37322607621124682694 T^{4} - 598055876188978 p^{7} T^{5} + 9605711756 p^{14} T^{6} - 114858 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 104952 T + 52207854272 T^{2} - 4176821606180040 T^{3} + \)\(12\!\cdots\!34\)\( T^{4} - 4176821606180040 p^{7} T^{5} + 52207854272 p^{14} T^{6} - 104952 p^{21} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 24976 T + 43688791996 T^{2} + 2739115850747216 T^{3} + \)\(14\!\cdots\!22\)\( T^{4} + 2739115850747216 p^{7} T^{5} + 43688791996 p^{14} T^{6} + 24976 p^{21} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 498856 T + 381977216092 T^{2} + 132519305800020216 T^{3} + \)\(54\!\cdots\!66\)\( T^{4} + 132519305800020216 p^{7} T^{5} + 381977216092 p^{14} T^{6} + 498856 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 17916 p T + 582543752696 T^{2} + 174072649937861684 T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + 174072649937861684 p^{7} T^{5} + 582543752696 p^{14} T^{6} + 17916 p^{22} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 201916 T + 337036845088 T^{2} + 169098883219921380 T^{3} + \)\(16\!\cdots\!34\)\( T^{4} + 169098883219921380 p^{7} T^{5} + 337036845088 p^{14} T^{6} - 201916 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 1995894 T + 2541886433372 T^{2} - 2356429844123408062 T^{3} + \)\(18\!\cdots\!58\)\( T^{4} - 2356429844123408062 p^{7} T^{5} + 2541886433372 p^{14} T^{6} - 1995894 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 929970 T + 696841347632 T^{2} + 410483380995330858 T^{3} - \)\(19\!\cdots\!26\)\( T^{4} + 410483380995330858 p^{7} T^{5} + 696841347632 p^{14} T^{6} - 929970 p^{21} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 1353156 T + 641417092700 T^{2} - 638087859099231076 T^{3} + \)\(35\!\cdots\!18\)\( T^{4} - 638087859099231076 p^{7} T^{5} + 641417092700 p^{14} T^{6} - 1353156 p^{21} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 3998774 T + 16202477625640 T^{2} + 36808547102293149042 T^{3} + \)\(80\!\cdots\!50\)\( T^{4} + 36808547102293149042 p^{7} T^{5} + 16202477625640 p^{14} T^{6} + 3998774 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 1722008 T + 22644499591660 T^{2} - 31607526991944054296 T^{3} + \)\(20\!\cdots\!06\)\( T^{4} - 31607526991944054296 p^{7} T^{5} + 22644499591660 p^{14} T^{6} - 1722008 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 5571858 T + 36809238259148 T^{2} - \)\(11\!\cdots\!98\)\( T^{3} + \)\(45\!\cdots\!94\)\( T^{4} - \)\(11\!\cdots\!98\)\( p^{7} T^{5} + 36809238259148 p^{14} T^{6} - 5571858 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 5600528 T + 40596655684732 T^{2} + \)\(18\!\cdots\!84\)\( T^{3} + \)\(65\!\cdots\!02\)\( T^{4} + \)\(18\!\cdots\!84\)\( p^{7} T^{5} + 40596655684732 p^{14} T^{6} + 5600528 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 7710226 T + 92181525864424 T^{2} - \)\(44\!\cdots\!70\)\( T^{3} + \)\(27\!\cdots\!30\)\( T^{4} - \)\(44\!\cdots\!70\)\( p^{7} T^{5} + 92181525864424 p^{14} T^{6} - 7710226 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 3431856 T + 43757629855388 T^{2} + 89436476395110295728 T^{3} + \)\(91\!\cdots\!66\)\( T^{4} + 89436476395110295728 p^{7} T^{5} + 43757629855388 p^{14} T^{6} + 3431856 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 4611528 T + 150275359100156 T^{2} - \)\(46\!\cdots\!64\)\( T^{3} + \)\(92\!\cdots\!06\)\( T^{4} - \)\(46\!\cdots\!64\)\( p^{7} T^{5} + 150275359100156 p^{14} T^{6} - 4611528 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 1401692 T + 169506769846084 T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!10\)\( T^{4} - \)\(11\!\cdots\!24\)\( p^{7} T^{5} + 169506769846084 p^{14} T^{6} - 1401692 p^{21} T^{7} + p^{28} T^{8} \)
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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

−7.06816308707770873715311340217, −6.94819930283901460451677213228, −6.72970149861705445629059560308, −6.45081495389205240486563253437, −6.25685315901166306378928972803, −5.94615580900540261699651067377, −5.59431861513585505724589557647, −5.22872414548983917656911588902, −5.01576969193143364892280461732, −4.63594095738563932512303784474, −4.28862470551017426555509362716, −4.17550547434237552667124028834, −3.59183622193585715975593938944, −3.34728228044571239174941400777, −3.16983346796733487431309523693, −3.12304030584119867231314258078, −2.54963505656958164396958939390, −2.31212456431414682320771515910, −2.15078942459788686348126876715, −1.85374366170036617439355192271, −1.49066482897220250708199718492, −1.44217453290855374541879106503, −0.74791317537543079436290975948, −0.49368913590802682179465343028, −0.42923036317711339109400682009, 0.42923036317711339109400682009, 0.49368913590802682179465343028, 0.74791317537543079436290975948, 1.44217453290855374541879106503, 1.49066482897220250708199718492, 1.85374366170036617439355192271, 2.15078942459788686348126876715, 2.31212456431414682320771515910, 2.54963505656958164396958939390, 3.12304030584119867231314258078, 3.16983346796733487431309523693, 3.34728228044571239174941400777, 3.59183622193585715975593938944, 4.17550547434237552667124028834, 4.28862470551017426555509362716, 4.63594095738563932512303784474, 5.01576969193143364892280461732, 5.22872414548983917656911588902, 5.59431861513585505724589557647, 5.94615580900540261699651067377, 6.25685315901166306378928972803, 6.45081495389205240486563253437, 6.72970149861705445629059560308, 6.94819930283901460451677213228, 7.06816308707770873715311340217

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