Properties

Label 8-384e4-1.1-c9e4-0-5
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

−7.45192201078529128199369587711, −6.89229481728761438724879068575, −6.80693671958811102945477677536, −6.63597994687138138091378095087, −6.42549973814888969048846300332, −5.81405610333415861105861096233, −5.56155717400560105135972829563, −5.40621802740334095729631829706, −5.33127281100290396541732582999, −4.65566021594373172377132275926, −4.34514906928017499444036197695, −4.26647665276004904128837885143, −4.12863721656855852786841137253, −3.76563561209575611149773539025, −3.41215309910467109080793863805, −3.34430304890932795882099883251, −3.15319610191790723436454812476, −2.92987832097011576000417611690, −2.35492764263575525859687751274, −2.26361173180376341522654435019, −2.24361489352617460362651716898, −1.46419906467736099514760605244, −1.45620570067432610814504426148, −1.21229496623749714675580093884, −0.954814099845870400005495442322, 0, 0, 0, 0, 0.954814099845870400005495442322, 1.21229496623749714675580093884, 1.45620570067432610814504426148, 1.46419906467736099514760605244, 2.24361489352617460362651716898, 2.26361173180376341522654435019, 2.35492764263575525859687751274, 2.92987832097011576000417611690, 3.15319610191790723436454812476, 3.34430304890932795882099883251, 3.41215309910467109080793863805, 3.76563561209575611149773539025, 4.12863721656855852786841137253, 4.26647665276004904128837885143, 4.34514906928017499444036197695, 4.65566021594373172377132275926, 5.33127281100290396541732582999, 5.40621802740334095729631829706, 5.56155717400560105135972829563, 5.81405610333415861105861096233, 6.42549973814888969048846300332, 6.63597994687138138091378095087, 6.80693671958811102945477677536, 6.89229481728761438724879068575, 7.45192201078529128199369587711

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