Properties

Label 8-363e4-1.1-c3e4-0-5
Degree $8$
Conductor $17363069361$
Sign $1$
Analytic cond. $210421.$
Root an. cond. $4.62792$
Motivic weight $3$
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  − 2-s + 12·3-s − 3·4-s + 14·5-s − 12·6-s + 20·7-s − 19·8-s + 90·9-s − 14·10-s − 36·12-s + 32·13-s − 20·14-s + 168·15-s + 49·16-s + 92·17-s − 90·18-s − 34·19-s − 42·20-s + 240·21-s − 26·23-s − 228·24-s + 15·25-s − 32·26-s + 540·27-s − 60·28-s + 174·29-s − 168·30-s + ⋯
L(s)  = 1  − 0.353·2-s + 2.30·3-s − 3/8·4-s + 1.25·5-s − 0.816·6-s + 1.07·7-s − 0.839·8-s + 10/3·9-s − 0.442·10-s − 0.866·12-s + 0.682·13-s − 0.381·14-s + 2.89·15-s + 0.765·16-s + 1.31·17-s − 1.17·18-s − 0.410·19-s − 0.469·20-s + 2.49·21-s − 0.235·23-s − 1.93·24-s + 3/25·25-s − 0.241·26-s + 3.84·27-s − 0.404·28-s + 1.11·29-s − 1.02·30-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(210421.\)
Root analytic conductor: \(4.62792\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(25.58370602\)
\(L(\frac12)\) \(\approx\) \(25.58370602\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
11 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + T + p^{2} T^{2} + 13 p T^{3} + p^{3} T^{4} + 13 p^{4} T^{5} + p^{8} T^{6} + p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 14 T + 181 T^{2} - 34 T^{3} - 3712 T^{4} - 34 p^{3} T^{5} + 181 p^{6} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 20 T + 745 T^{2} - 250 p T^{3} + 151460 T^{4} - 250 p^{4} T^{5} + 745 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 32 T + 6283 T^{2} - 172324 T^{3} + 19678376 T^{4} - 172324 p^{3} T^{5} + 6283 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 92 T + 10015 T^{2} - 1708 p^{2} T^{3} + 48261836 T^{4} - 1708 p^{5} T^{5} + 10015 p^{6} T^{6} - 92 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 34 T + 7789 T^{2} + 491934 T^{3} + 100192812 T^{4} + 491934 p^{3} T^{5} + 7789 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 26 T + 5800 T^{2} + 628762 T^{3} + 272602574 T^{4} + 628762 p^{3} T^{5} + 5800 p^{6} T^{6} + 26 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 6 p T + 89405 T^{2} - 10079334 T^{3} + 3059530596 T^{4} - 10079334 p^{3} T^{5} + 89405 p^{6} T^{6} - 6 p^{10} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 422 T + 121621 T^{2} - 813586 p T^{3} + 4715438036 T^{4} - 813586 p^{4} T^{5} + 121621 p^{6} T^{6} - 422 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 14 p T + 222196 T^{2} - 69969108 T^{3} + 17042241381 T^{4} - 69969108 p^{3} T^{5} + 222196 p^{6} T^{6} - 14 p^{10} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 428 T + 281623 T^{2} - 72996448 T^{3} + 27843065288 T^{4} - 72996448 p^{3} T^{5} + 281623 p^{6} T^{6} - 428 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 550 T + 356860 T^{2} + 122234838 T^{3} + 43628494230 T^{4} + 122234838 p^{3} T^{5} + 356860 p^{6} T^{6} + 550 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 556 T + 464992 T^{2} - 159161852 T^{3} + 73095550526 T^{4} - 159161852 p^{3} T^{5} + 464992 p^{6} T^{6} - 556 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 882 T + 753641 T^{2} - 356933250 T^{3} + 170508810204 T^{4} - 356933250 p^{3} T^{5} + 753641 p^{6} T^{6} - 882 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 158 T + 455704 T^{2} - 10644746 T^{3} + 99834013694 T^{4} - 10644746 p^{3} T^{5} + 455704 p^{6} T^{6} + 158 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 290 T + 780649 T^{2} - 160512114 T^{3} + 249052940940 T^{4} - 160512114 p^{3} T^{5} + 780649 p^{6} T^{6} - 290 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 992 T + 1325281 T^{2} - 870459746 T^{3} + 613021090972 T^{4} - 870459746 p^{3} T^{5} + 1325281 p^{6} T^{6} - 992 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 42 T + 203372 T^{2} - 102129390 T^{3} - 39343991994 T^{4} - 102129390 p^{3} T^{5} + 203372 p^{6} T^{6} + 42 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1274 T + 1771225 T^{2} - 1360918918 T^{3} + 1080945673244 T^{4} - 1360918918 p^{3} T^{5} + 1771225 p^{6} T^{6} - 1274 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 362 T + 1421725 T^{2} - 271468310 T^{3} + 902647752772 T^{4} - 271468310 p^{3} T^{5} + 1421725 p^{6} T^{6} - 362 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1500 T + 2217920 T^{2} - 1968382332 T^{3} + 1823815616910 T^{4} - 1968382332 p^{3} T^{5} + 2217920 p^{6} T^{6} - 1500 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 1428 T + 2335535 T^{2} - 2459628540 T^{3} + 2471453298972 T^{4} - 2459628540 p^{3} T^{5} + 2335535 p^{6} T^{6} - 1428 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 1052 T + 2038366 T^{2} - 1884713816 T^{3} + 2895539158591 T^{4} - 1884713816 p^{3} T^{5} + 2038366 p^{6} T^{6} - 1052 p^{9} T^{7} + p^{12} T^{8} \)
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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

−8.040018462962733461985957115522, −7.64874480450264325862200703321, −7.63643024399752479568047197688, −7.33839364275015520573928286484, −6.69791578370613572341247882103, −6.66047822580691217834369004591, −6.30839304734958352742320078045, −6.04144296615964825300752378107, −5.96948621316237223497226329494, −5.49511072594376716864572374396, −5.10743244778445591163025299232, −4.90131527281488048692421647305, −4.70253144711563399570300509452, −4.22342993459894077008416425004, −3.90997729529186602117119618084, −3.71920781050341466681657428245, −3.25670937142769234473578939061, −3.18336058765189940281318337804, −2.56931066461482979505730390307, −2.41189745023876140237546714382, −2.04851045105112255784558298752, −1.98624066776787347617820604462, −1.06000521128116758827770847633, −0.910446496688850171119099671549, −0.825568935356703110791253102644, 0.825568935356703110791253102644, 0.910446496688850171119099671549, 1.06000521128116758827770847633, 1.98624066776787347617820604462, 2.04851045105112255784558298752, 2.41189745023876140237546714382, 2.56931066461482979505730390307, 3.18336058765189940281318337804, 3.25670937142769234473578939061, 3.71920781050341466681657428245, 3.90997729529186602117119618084, 4.22342993459894077008416425004, 4.70253144711563399570300509452, 4.90131527281488048692421647305, 5.10743244778445591163025299232, 5.49511072594376716864572374396, 5.96948621316237223497226329494, 6.04144296615964825300752378107, 6.30839304734958352742320078045, 6.66047822580691217834369004591, 6.69791578370613572341247882103, 7.33839364275015520573928286484, 7.63643024399752479568047197688, 7.64874480450264325862200703321, 8.040018462962733461985957115522

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