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Properties

Label 8-273e4-1.1-c3e4-0-1
Degree $8$
Conductor $5554571841$
Sign $1$
Analytic cond. $67315.4$
Root an. cond. $4.01341$
Motivic weight $3$
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  − 3·2-s + 12·3-s − 4·4-s − 24·5-s − 36·6-s − 28·7-s + 16·8-s + 90·9-s + 72·10-s + 8·11-s − 48·12-s − 52·13-s + 84·14-s − 288·15-s − 59·16-s − 6·17-s − 270·18-s − 332·19-s + 96·20-s − 336·21-s − 24·22-s + 6·23-s + 192·24-s + 32·25-s + 156·26-s + 540·27-s + 112·28-s + ⋯
L(s)  = 1  − 1.06·2-s + 2.30·3-s − 1/2·4-s − 2.14·5-s − 2.44·6-s − 1.51·7-s + 0.707·8-s + 10/3·9-s + 2.27·10-s + 0.219·11-s − 1.15·12-s − 1.10·13-s + 1.60·14-s − 4.95·15-s − 0.921·16-s − 0.0856·17-s − 3.53·18-s − 4.00·19-s + 1.07·20-s − 3.49·21-s − 0.232·22-s + 0.0543·23-s + 1.63·24-s + 0.255·25-s + 1.17·26-s + 3.84·27-s + 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(67315.4\)
Root analytic conductor: \(4.01341\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
7$C_1$ \( ( 1 + p T )^{4} \)
13$C_1$ \( ( 1 + p T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + 3 T + 13 T^{2} + 35 T^{3} + 21 p^{3} T^{4} + 35 p^{3} T^{5} + 13 p^{6} T^{6} + 3 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 24 T + 544 T^{2} + 7496 T^{3} + 102638 T^{4} + 7496 p^{3} T^{5} + 544 p^{6} T^{6} + 24 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 8 T + 1696 T^{2} - 42872 T^{3} + 3216494 T^{4} - 42872 p^{3} T^{5} + 1696 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 6 T + 4832 T^{2} + 260554 T^{3} + 14796958 T^{4} + 260554 p^{3} T^{5} + 4832 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 332 T + 67372 T^{2} + 8895564 T^{3} + 868684246 T^{4} + 8895564 p^{3} T^{5} + 67372 p^{6} T^{6} + 332 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 6 T + 14956 T^{2} - 1737038 T^{3} + 96199622 T^{4} - 1737038 p^{3} T^{5} + 14956 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 8 T + 53452 T^{2} + 76296 p T^{3} + 1527806806 T^{4} + 76296 p^{4} T^{5} + 53452 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 184 T + 59820 T^{2} + 10792408 T^{3} + 2674049830 T^{4} + 10792408 p^{3} T^{5} + 59820 p^{6} T^{6} + 184 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 244 T + 166548 T^{2} + 25776796 T^{3} + 11264423526 T^{4} + 25776796 p^{3} T^{5} + 166548 p^{6} T^{6} + 244 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 296 T + 248784 T^{2} + 49944472 T^{3} + 24206939518 T^{4} + 49944472 p^{3} T^{5} + 248784 p^{6} T^{6} + 296 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 224 T + 63820 T^{2} + 29609184 T^{3} + 13135501174 T^{4} + 29609184 p^{3} T^{5} + 63820 p^{6} T^{6} + 224 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 750 T + 411408 T^{2} + 155350766 T^{3} + 53955126366 T^{4} + 155350766 p^{3} T^{5} + 411408 p^{6} T^{6} + 750 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 680 T + 152252 T^{2} + 1051016 T^{3} - 1284313962 T^{4} + 1051016 p^{3} T^{5} + 152252 p^{6} T^{6} - 680 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 334 T + 599296 T^{2} + 188815830 T^{3} + 170585131950 T^{4} + 188815830 p^{3} T^{5} + 599296 p^{6} T^{6} + 334 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 1212 T + 844308 T^{2} + 344747284 T^{3} + 146786940422 T^{4} + 344747284 p^{3} T^{5} + 844308 p^{6} T^{6} + 1212 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 388 T + 473180 T^{2} - 32608636 T^{3} + 87702717110 T^{4} - 32608636 p^{3} T^{5} + 473180 p^{6} T^{6} + 388 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 544 T + 1296304 T^{2} + 479697456 T^{3} + 659661843454 T^{4} + 479697456 p^{3} T^{5} + 1296304 p^{6} T^{6} + 544 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 1572 T + 2388516 T^{2} + 2000366716 T^{3} + 1565312286230 T^{4} + 2000366716 p^{3} T^{5} + 2388516 p^{6} T^{6} + 1572 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 464 T + 1266428 T^{2} - 712788880 T^{3} + 9700283354 p T^{4} - 712788880 p^{3} T^{5} + 1266428 p^{6} T^{6} - 464 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 3158 T + 5826576 T^{2} + 7035637486 T^{3} + 6246698059822 T^{4} + 7035637486 p^{3} T^{5} + 5826576 p^{6} T^{6} + 3158 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 1156 T + 2174632 T^{2} - 2244619372 T^{3} + 2124576231470 T^{4} - 2244619372 p^{3} T^{5} + 2174632 p^{6} T^{6} - 1156 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 1928 T + 3272412 T^{2} + 4273091768 T^{3} + 4508754061638 T^{4} + 4273091768 p^{3} T^{5} + 3272412 p^{6} T^{6} + 1928 p^{9} T^{7} + p^{12} T^{8} \)
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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

−8.894448241411533521841286751869, −8.418175648089568551351309984985, −8.280357360531041760961439718811, −8.145801672801208426491187082817, −7.938991252312899615929166571797, −7.51291968985861468054757317096, −7.31599262995772135531954771669, −7.04020147905273403078565689393, −6.90750867273500483200624116186, −6.47638331546045506963207569932, −6.42488185718620005086346197460, −5.87374423233233219471906682620, −5.62564738575194390753396363981, −4.80557039459584497560570849717, −4.51914457513740680903559553780, −4.40618038424578347973170281322, −4.35559576565633025827202971516, −3.78790407769704292634900161188, −3.61677854802155546392833856272, −3.46386040672031235450103384523, −2.86222383354512083646048169225, −2.69028018268842629633239632605, −2.25161210551191020323912068774, −1.76408399641155180673019447581, −1.50284943508423719117319968989, 0, 0, 0, 0, 1.50284943508423719117319968989, 1.76408399641155180673019447581, 2.25161210551191020323912068774, 2.69028018268842629633239632605, 2.86222383354512083646048169225, 3.46386040672031235450103384523, 3.61677854802155546392833856272, 3.78790407769704292634900161188, 4.35559576565633025827202971516, 4.40618038424578347973170281322, 4.51914457513740680903559553780, 4.80557039459584497560570849717, 5.62564738575194390753396363981, 5.87374423233233219471906682620, 6.42488185718620005086346197460, 6.47638331546045506963207569932, 6.90750867273500483200624116186, 7.04020147905273403078565689393, 7.31599262995772135531954771669, 7.51291968985861468054757317096, 7.938991252312899615929166571797, 8.145801672801208426491187082817, 8.280357360531041760961439718811, 8.418175648089568551351309984985, 8.894448241411533521841286751869

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