Properties

Label 8-384e4-1.1-c9e4-0-4
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $1.52994\times 10^{9}$
Root an. cond. $14.0632$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 324·3-s + 1.72e3·5-s − 4.84e3·7-s + 6.56e4·9-s − 1.58e4·11-s + 8.24e4·13-s − 5.59e5·15-s + 1.65e5·17-s − 5.39e5·19-s + 1.56e6·21-s − 7.29e5·23-s − 2.30e6·25-s − 1.06e7·27-s + 1.85e6·29-s − 3.19e6·31-s + 5.12e6·33-s − 8.36e6·35-s + 4.18e6·37-s − 2.67e7·39-s + 2.27e5·41-s − 1.60e6·43-s + 1.13e8·45-s − 1.80e7·47-s − 4.01e7·49-s − 5.37e7·51-s + 2.94e7·53-s − 2.73e7·55-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.23·5-s − 0.761·7-s + 10/3·9-s − 0.325·11-s + 0.800·13-s − 2.85·15-s + 0.481·17-s − 0.950·19-s + 1.75·21-s − 0.543·23-s − 1.17·25-s − 3.84·27-s + 0.485·29-s − 0.621·31-s + 0.752·33-s − 0.942·35-s + 0.367·37-s − 1.84·39-s + 0.0125·41-s − 0.0713·43-s + 4.12·45-s − 0.539·47-s − 0.994·49-s − 1.11·51-s + 0.512·53-s − 0.402·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.52994\times 10^{9}\)
Root analytic conductor: \(14.0632\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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$C_1$ \( ( 1 + p^{4} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 1728 T + 5286836 T^{2} - 1344360512 p T^{3} + 585667235526 p^{2} T^{4} - 1344360512 p^{10} T^{5} + 5286836 p^{18} T^{6} - 1728 p^{27} T^{7} + p^{36} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 4840 T + 9079372 p T^{2} + 9122761384 p^{2} T^{3} + 9758964239914 p^{3} T^{4} + 9122761384 p^{11} T^{5} + 9079372 p^{19} T^{6} + 4840 p^{27} T^{7} + p^{36} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 15824 T + 3272030348 T^{2} + 110832700706512 T^{3} + 11168899925921994134 T^{4} + 110832700706512 p^{9} T^{5} + 3272030348 p^{18} T^{6} + 15824 p^{27} T^{7} + p^{36} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 82440 T + 32507843788 T^{2} - 2806427918898776 T^{3} + \)\(46\!\cdots\!74\)\( T^{4} - 2806427918898776 p^{9} T^{5} + 32507843788 p^{18} T^{6} - 82440 p^{27} T^{7} + p^{36} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 165912 T + 402742640540 T^{2} - 57573640941296168 T^{3} + \)\(68\!\cdots\!90\)\( T^{4} - 57573640941296168 p^{9} T^{5} + 402742640540 p^{18} T^{6} - 165912 p^{27} T^{7} + p^{36} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 28416 p T + 878148739660 T^{2} + 411395194240546048 T^{3} + \)\(35\!\cdots\!02\)\( T^{4} + 411395194240546048 p^{9} T^{5} + 878148739660 p^{18} T^{6} + 28416 p^{28} T^{7} + p^{36} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 729680 T + 3870147116348 T^{2} + 3090843717350364304 T^{3} + \)\(76\!\cdots\!54\)\( T^{4} + 3090843717350364304 p^{9} T^{5} + 3870147116348 p^{18} T^{6} + 729680 p^{27} T^{7} + p^{36} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 1850864 T + 51896421352052 T^{2} - 76884864756743058576 T^{3} + \)\(10\!\cdots\!42\)\( T^{4} - 76884864756743058576 p^{9} T^{5} + 51896421352052 p^{18} T^{6} - 1850864 p^{27} T^{7} + p^{36} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 103160 p T + 80766069413044 T^{2} + \)\(19\!\cdots\!92\)\( T^{3} + \)\(28\!\cdots\!74\)\( T^{4} + \)\(19\!\cdots\!92\)\( p^{9} T^{5} + 80766069413044 p^{18} T^{6} + 103160 p^{28} T^{7} + p^{36} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 4187992 T + 316603511258092 T^{2} - \)\(22\!\cdots\!72\)\( T^{3} + \)\(48\!\cdots\!50\)\( T^{4} - \)\(22\!\cdots\!72\)\( p^{9} T^{5} + 316603511258092 p^{18} T^{6} - 4187992 p^{27} T^{7} + p^{36} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 227704 T + 815482715263292 T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!46\)\( T^{4} - \)\(24\!\cdots\!80\)\( p^{9} T^{5} + 815482715263292 p^{18} T^{6} - 227704 p^{27} T^{7} + p^{36} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 1600352 T + 1039425217732396 T^{2} - \)\(25\!\cdots\!40\)\( T^{3} + \)\(54\!\cdots\!14\)\( T^{4} - \)\(25\!\cdots\!40\)\( p^{9} T^{5} + 1039425217732396 p^{18} T^{6} + 1600352 p^{27} T^{7} + p^{36} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 18053904 T + 898850271089564 T^{2} + \)\(31\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!54\)\( T^{4} + \)\(31\!\cdots\!80\)\( p^{9} T^{5} + 898850271089564 p^{18} T^{6} + 18053904 p^{27} T^{7} + p^{36} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 29418288 T + 8246475311053268 T^{2} - \)\(75\!\cdots\!52\)\( T^{3} + \)\(31\!\cdots\!10\)\( T^{4} - \)\(75\!\cdots\!52\)\( p^{9} T^{5} + 8246475311053268 p^{18} T^{6} - 29418288 p^{27} T^{7} + p^{36} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 38300048 T + 12480247782213068 T^{2} + \)\(64\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!66\)\( T^{4} + \)\(64\!\cdots\!60\)\( p^{9} T^{5} + 12480247782213068 p^{18} T^{6} + 38300048 p^{27} T^{7} + p^{36} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 99764648 T + 35070113945959756 T^{2} + \)\(25\!\cdots\!08\)\( T^{3} + \)\(53\!\cdots\!54\)\( T^{4} + \)\(25\!\cdots\!08\)\( p^{9} T^{5} + 35070113945959756 p^{18} T^{6} + 99764648 p^{27} T^{7} + p^{36} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 183717008 T + 94982362218124012 T^{2} - \)\(13\!\cdots\!36\)\( T^{3} + \)\(56\!\cdots\!98\)\( p T^{4} - \)\(13\!\cdots\!36\)\( p^{9} T^{5} + 94982362218124012 p^{18} T^{6} - 183717008 p^{27} T^{7} + p^{36} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 181868080 T + 187559006549429756 T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} + \)\(24\!\cdots\!80\)\( p^{9} T^{5} + 187559006549429756 p^{18} T^{6} + 181868080 p^{27} T^{7} + p^{36} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 254539160 T + 208692556022844412 T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(17\!\cdots\!82\)\( T^{4} - \)\(39\!\cdots\!64\)\( p^{9} T^{5} + 208692556022844412 p^{18} T^{6} - 254539160 p^{27} T^{7} + p^{36} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 831578184 T + 541079473633225972 T^{2} + \)\(23\!\cdots\!64\)\( T^{3} + \)\(93\!\cdots\!90\)\( T^{4} + \)\(23\!\cdots\!64\)\( p^{9} T^{5} + 541079473633225972 p^{18} T^{6} + 831578184 p^{27} T^{7} + p^{36} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 687923952 T + 569668186782399404 T^{2} - \)\(18\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!50\)\( T^{4} - \)\(18\!\cdots\!04\)\( p^{9} T^{5} + 569668186782399404 p^{18} T^{6} - 687923952 p^{27} T^{7} + p^{36} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 627627272 T + 1209510335623979132 T^{2} - \)\(64\!\cdots\!44\)\( T^{3} + \)\(60\!\cdots\!14\)\( T^{4} - \)\(64\!\cdots\!44\)\( p^{9} T^{5} + 1209510335623979132 p^{18} T^{6} - 627627272 p^{27} T^{7} + p^{36} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 889385880 T + 757736193785874268 T^{2} + \)\(68\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!34\)\( T^{4} + \)\(68\!\cdots\!80\)\( p^{9} T^{5} + 757736193785874268 p^{18} T^{6} + 889385880 p^{27} T^{7} + p^{36} T^{8} \)
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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

−7.07927852422876537833622015792, −6.60482062508142432657615452707, −6.54236793459762773333085607630, −6.31198422905612158357922018632, −6.13163048679435976081795541304, −5.95203174342194529914137030036, −5.69795532451779092923244917892, −5.51980552153942977745741726276, −5.50584188455827829707867648654, −4.85130745201472933222742481162, −4.60364643630617710109456710164, −4.54329600427176312110668771429, −4.52061136215770887389803068766, −3.74358436155851938490179994395, −3.52776463430214895631583154263, −3.41346325040666160447515432278, −3.39185306807798248091796413595, −2.46234406386209354725024368350, −2.32992353924345293166665294498, −2.06880944479956918880793750274, −2.06029034292179357246057522206, −1.35345390875879127736044968231, −1.21623986970956198670954451340, −1.10264839319436809075677086964, −0.823063173700572891650597203094, 0, 0, 0, 0, 0.823063173700572891650597203094, 1.10264839319436809075677086964, 1.21623986970956198670954451340, 1.35345390875879127736044968231, 2.06029034292179357246057522206, 2.06880944479956918880793750274, 2.32992353924345293166665294498, 2.46234406386209354725024368350, 3.39185306807798248091796413595, 3.41346325040666160447515432278, 3.52776463430214895631583154263, 3.74358436155851938490179994395, 4.52061136215770887389803068766, 4.54329600427176312110668771429, 4.60364643630617710109456710164, 4.85130745201472933222742481162, 5.50584188455827829707867648654, 5.51980552153942977745741726276, 5.69795532451779092923244917892, 5.95203174342194529914137030036, 6.13163048679435976081795541304, 6.31198422905612158357922018632, 6.54236793459762773333085607630, 6.60482062508142432657615452707, 7.07927852422876537833622015792

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