Properties

Label 8-363e4-1.1-c9e4-0-0
Degree $8$
Conductor $17363069361$
Sign $1$
Analytic cond. $1.22173\times 10^{9}$
Root an. cond. $13.6732$
Motivic weight $9$
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  − 19·2-s − 324·3-s + 125·4-s + 1.13e3·5-s + 6.15e3·6-s − 8.12e3·7-s + 5.53e3·8-s + 6.56e4·9-s − 2.16e4·10-s − 4.05e4·12-s − 2.30e5·13-s + 1.54e5·14-s − 3.68e5·15-s + 8.88e4·16-s + 116·17-s − 1.24e6·18-s − 1.17e6·19-s + 1.42e5·20-s + 2.63e6·21-s + 3.17e5·23-s − 1.79e6·24-s − 2.30e6·25-s + 4.38e6·26-s − 1.06e7·27-s − 1.01e6·28-s − 2.52e5·29-s + 7.00e6·30-s + ⋯
L(s)  = 1  − 0.839·2-s − 2.30·3-s + 0.244·4-s + 0.814·5-s + 1.93·6-s − 1.27·7-s + 0.477·8-s + 10/3·9-s − 0.683·10-s − 0.563·12-s − 2.24·13-s + 1.07·14-s − 1.88·15-s + 0.338·16-s + 0.000336·17-s − 2.79·18-s − 2.07·19-s + 0.198·20-s + 2.95·21-s + 0.236·23-s − 1.10·24-s − 1.18·25-s + 1.88·26-s − 3.84·27-s − 0.312·28-s − 0.0662·29-s + 1.57·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.22173\times 10^{9}\)
Root analytic conductor: \(13.6732\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(0.009763079483\)
\(L(\frac12)\) \(\approx\) \(0.009763079483\)
\(L(\frac{11}{2})\) 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)$
bad3$C_1$ \( ( 1 + p^{4} T )^{4} \)
11 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + 19 T + 59 p^{2} T^{2} - 857 p^{2} T^{3} - 18041 p^{4} T^{4} - 857 p^{11} T^{5} + 59 p^{20} T^{6} + 19 p^{27} T^{7} + p^{36} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 1138 T + 720616 p T^{2} - 124657942 p^{2} T^{3} + 47006730518 p^{3} T^{4} - 124657942 p^{11} T^{5} + 720616 p^{19} T^{6} - 1138 p^{27} T^{7} + p^{36} T^{8} \)
7$C_2 \wr S_4$ \( 1 + 8122 T + 139046428 T^{2} + 106622327566 p T^{3} + 157611327816454 p^{2} T^{4} + 106622327566 p^{10} T^{5} + 139046428 p^{18} T^{6} + 8122 p^{27} T^{7} + p^{36} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 230998 T + 51524355232 T^{2} + 6860116534790914 T^{3} + \)\(87\!\cdots\!50\)\( T^{4} + 6860116534790914 p^{9} T^{5} + 51524355232 p^{18} T^{6} + 230998 p^{27} T^{7} + p^{36} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 116 T + 197930548676 T^{2} + 43933337930841652 T^{3} + \)\(22\!\cdots\!70\)\( T^{4} + 43933337930841652 p^{9} T^{5} + 197930548676 p^{18} T^{6} - 116 p^{27} T^{7} + p^{36} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 1179324 T + 78430932868 p T^{2} + 1079697791011100508 T^{3} + \)\(74\!\cdots\!54\)\( T^{4} + 1079697791011100508 p^{9} T^{5} + 78430932868 p^{19} T^{6} + 1179324 p^{27} T^{7} + p^{36} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 317378 T + 4317638946644 T^{2} + 1401830589166572214 T^{3} + \)\(81\!\cdots\!10\)\( T^{4} + 1401830589166572214 p^{9} T^{5} + 4317638946644 p^{18} T^{6} - 317378 p^{27} T^{7} + p^{36} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 252516 T + 24768563521460 T^{2} - 3090837853393370388 T^{3} + \)\(53\!\cdots\!82\)\( T^{4} - 3090837853393370388 p^{9} T^{5} + 24768563521460 p^{18} T^{6} + 252516 p^{27} T^{7} + p^{36} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 7812776 T + 26020457637436 T^{2} - 65600974865495185144 T^{3} - \)\(80\!\cdots\!38\)\( T^{4} - 65600974865495185144 p^{9} T^{5} + 26020457637436 p^{18} T^{6} + 7812776 p^{27} T^{7} + p^{36} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 13990056 T + 212347108746508 T^{2} - \)\(29\!\cdots\!32\)\( T^{3} + \)\(50\!\cdots\!50\)\( T^{4} - \)\(29\!\cdots\!32\)\( p^{9} T^{5} + 212347108746508 p^{18} T^{6} - 13990056 p^{27} T^{7} + p^{36} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 68389448 T + 1836656099234300 T^{2} - \)\(23\!\cdots\!64\)\( T^{3} + \)\(23\!\cdots\!62\)\( T^{4} - \)\(23\!\cdots\!64\)\( p^{9} T^{5} + 1836656099234300 p^{18} T^{6} - 68389448 p^{27} T^{7} + p^{36} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 6207552 T + 1679432704788460 T^{2} + \)\(82\!\cdots\!88\)\( T^{3} + \)\(12\!\cdots\!62\)\( T^{4} + \)\(82\!\cdots\!88\)\( p^{9} T^{5} + 1679432704788460 p^{18} T^{6} + 6207552 p^{27} T^{7} + p^{36} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 30703042 T + 3420877066678388 T^{2} + \)\(64\!\cdots\!70\)\( T^{3} + \)\(48\!\cdots\!10\)\( T^{4} + \)\(64\!\cdots\!70\)\( p^{9} T^{5} + 3420877066678388 p^{18} T^{6} + 30703042 p^{27} T^{7} + p^{36} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 41285850 T + 5829902498169608 T^{2} - \)\(26\!\cdots\!26\)\( T^{3} + \)\(30\!\cdots\!74\)\( T^{4} - \)\(26\!\cdots\!26\)\( p^{9} T^{5} + 5829902498169608 p^{18} T^{6} - 41285850 p^{27} T^{7} + p^{36} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 132805180 T + 32077497142785404 T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + \)\(41\!\cdots\!46\)\( T^{4} + \)\(29\!\cdots\!80\)\( p^{9} T^{5} + 32077497142785404 p^{18} T^{6} + 132805180 p^{27} T^{7} + p^{36} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 276571338 T + 72924677895261208 T^{2} + \)\(10\!\cdots\!42\)\( T^{3} + \)\(14\!\cdots\!30\)\( T^{4} + \)\(10\!\cdots\!42\)\( p^{9} T^{5} + 72924677895261208 p^{18} T^{6} + 276571338 p^{27} T^{7} + p^{36} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 210077288 T + 95220713160869068 T^{2} - \)\(11\!\cdots\!88\)\( T^{3} + \)\(34\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!88\)\( p^{9} T^{5} + 95220713160869068 p^{18} T^{6} - 210077288 p^{27} T^{7} + p^{36} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 270008586 T + 151484134438028516 T^{2} - \)\(37\!\cdots\!38\)\( T^{3} + \)\(96\!\cdots\!70\)\( T^{4} - \)\(37\!\cdots\!38\)\( p^{9} T^{5} + 151484134438028516 p^{18} T^{6} - 270008586 p^{27} T^{7} + p^{36} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 314963656 T + 7118228466007228 T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(72\!\cdots\!82\)\( T^{4} + \)\(10\!\cdots\!72\)\( p^{9} T^{5} + 7118228466007228 p^{18} T^{6} + 314963656 p^{27} T^{7} + p^{36} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1553310662 T + 1310075286856762780 T^{2} + \)\(73\!\cdots\!10\)\( T^{3} + \)\(29\!\cdots\!26\)\( T^{4} + \)\(73\!\cdots\!10\)\( p^{9} T^{5} + 1310075286856762780 p^{18} T^{6} + 1553310662 p^{27} T^{7} + p^{36} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 500616048 T + 691323778471349228 T^{2} + \)\(23\!\cdots\!92\)\( T^{3} + \)\(18\!\cdots\!74\)\( T^{4} + \)\(23\!\cdots\!92\)\( p^{9} T^{5} + 691323778471349228 p^{18} T^{6} + 500616048 p^{27} T^{7} + p^{36} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1406681304 T + 1532137378557578684 T^{2} + \)\(12\!\cdots\!96\)\( T^{3} + \)\(85\!\cdots\!18\)\( T^{4} + \)\(12\!\cdots\!96\)\( p^{9} T^{5} + 1532137378557578684 p^{18} T^{6} + 1406681304 p^{27} T^{7} + p^{36} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 2126336996 T + 4332534312329313796 T^{2} - \)\(50\!\cdots\!48\)\( T^{3} + \)\(54\!\cdots\!30\)\( T^{4} - \)\(50\!\cdots\!48\)\( p^{9} T^{5} + 4332534312329313796 p^{18} T^{6} - 2126336996 p^{27} T^{7} + p^{36} T^{8} \)
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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

−6.51009403217775663924467635720, −6.35431185266643589749542412667, −6.34678141479847310599207028922, −5.89632338456251211940768086145, −5.85613413392240341024370634691, −5.69212607622661035896281454741, −5.26512026465248711037011312279, −4.95870969168499021079811484895, −4.92838070765005386361029613140, −4.30959687700080595385876117929, −4.28200267795667318112561310274, −4.18070082031612945070265649305, −3.79731521464729643537822305841, −3.32133927659999696106341761477, −2.86443904387179614024368525336, −2.73054666857202443812199588868, −2.38486838153370444336038512625, −2.17509495216485557953346509896, −1.64655990159612116676982963219, −1.50423117135605893575297596129, −1.36602832922054265783674917319, −1.00381495709335753348466347715, −0.45868962980851588837791108778, −0.17968157028448797278430023222, −0.05663087810793102111497272456, 0.05663087810793102111497272456, 0.17968157028448797278430023222, 0.45868962980851588837791108778, 1.00381495709335753348466347715, 1.36602832922054265783674917319, 1.50423117135605893575297596129, 1.64655990159612116676982963219, 2.17509495216485557953346509896, 2.38486838153370444336038512625, 2.73054666857202443812199588868, 2.86443904387179614024368525336, 3.32133927659999696106341761477, 3.79731521464729643537822305841, 4.18070082031612945070265649305, 4.28200267795667318112561310274, 4.30959687700080595385876117929, 4.92838070765005386361029613140, 4.95870969168499021079811484895, 5.26512026465248711037011312279, 5.69212607622661035896281454741, 5.85613413392240341024370634691, 5.89632338456251211940768086145, 6.34678141479847310599207028922, 6.35431185266643589749542412667, 6.51009403217775663924467635720

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