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Properties

Label 8-1925e4-1.1-c3e4-0-1
Degree $8$
Conductor $1.373\times 10^{13}$
Sign $1$
Analytic cond. $1.66412\times 10^{8}$
Root an. cond. $10.6573$
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  + 2·2-s − 14·3-s − 4-s − 28·6-s + 28·7-s − 4·8-s + 82·9-s − 44·11-s + 14·12-s − 58·13-s + 56·14-s − 51·16-s − 4·17-s + 164·18-s + 258·19-s − 392·21-s − 88·22-s − 8·23-s + 56·24-s − 116·26-s − 250·27-s − 28·28-s − 396·29-s − 56·31-s − 154·32-s + 616·33-s − 8·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.69·3-s − 1/8·4-s − 1.90·6-s + 1.51·7-s − 0.176·8-s + 3.03·9-s − 1.20·11-s + 0.336·12-s − 1.23·13-s + 1.06·14-s − 0.796·16-s − 0.0570·17-s + 2.14·18-s + 3.11·19-s − 4.07·21-s − 0.852·22-s − 0.0725·23-s + 0.476·24-s − 0.874·26-s − 1.78·27-s − 0.188·28-s − 2.53·29-s − 0.324·31-s − 0.850·32-s + 3.24·33-s − 0.0403·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1.66412\times 10^{8}\)
Root analytic conductor: \(10.6573\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{8} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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)$
bad5 \( 1 \)
7$C_1$ \( ( 1 - p T )^{4} \)
11$C_1$ \( ( 1 + p T )^{4} \)
good2$C_2 \wr S_4$ \( 1 - p T + 5 T^{2} - p^{3} T^{3} + p^{6} T^{4} - p^{6} T^{5} + 5 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
3$C_2 \wr S_4$ \( 1 + 14 T + 38 p T^{2} + 698 T^{3} + 3682 T^{4} + 698 p^{3} T^{5} + 38 p^{7} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 58 T + 3862 T^{2} + 39850 T^{3} + 2368354 T^{4} + 39850 p^{3} T^{5} + 3862 p^{6} T^{6} + 58 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 4 T + 13466 T^{2} - 198500 T^{3} + 81336754 T^{4} - 198500 p^{3} T^{5} + 13466 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 258 T + 2296 p T^{2} - 5147154 T^{3} + 489918366 T^{4} - 5147154 p^{3} T^{5} + 2296 p^{7} T^{6} - 258 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 8 T + 26276 T^{2} - 1258456 T^{3} + 325878550 T^{4} - 1258456 p^{3} T^{5} + 26276 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 396 T + 138428 T^{2} + 30598020 T^{3} + 5584930806 T^{4} + 30598020 p^{3} T^{5} + 138428 p^{6} T^{6} + 396 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 56 T + 87274 T^{2} + 3513992 T^{3} + 3413701858 T^{4} + 3513992 p^{3} T^{5} + 87274 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 84 T + 139096 T^{2} + 12117036 T^{3} + 8970963870 T^{4} + 12117036 p^{3} T^{5} + 139096 p^{6} T^{6} + 84 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 52 T + 191978 T^{2} + 4000964 T^{3} + 16303189330 T^{4} + 4000964 p^{3} T^{5} + 191978 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 408 T + 227224 T^{2} + 65489736 T^{3} + 24699468414 T^{4} + 65489736 p^{3} T^{5} + 227224 p^{6} T^{6} + 408 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 8 T + 293642 T^{2} + 14787896 T^{3} + 39096224482 T^{4} + 14787896 p^{3} T^{5} + 293642 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 624 T + 731456 T^{2} + 291124560 T^{3} + 173869150638 T^{4} + 291124560 p^{3} T^{5} + 731456 p^{6} T^{6} + 624 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 238 T + 320090 T^{2} - 3740918 T^{3} + 64718281666 T^{4} - 3740918 p^{3} T^{5} + 320090 p^{6} T^{6} + 238 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 162 T + 320206 T^{2} - 56573262 T^{3} + 48989538114 T^{4} - 56573262 p^{3} T^{5} + 320206 p^{6} T^{6} + 162 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 20 p T + 1030948 T^{2} + 550745180 T^{3} + 298359795814 T^{4} + 550745180 p^{3} T^{5} + 1030948 p^{6} T^{6} + 20 p^{10} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 1788 T + 2193428 T^{2} - 1809946572 T^{3} + 1240921367286 T^{4} - 1809946572 p^{3} T^{5} + 2193428 p^{6} T^{6} - 1788 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 1456 T + 1167226 T^{2} + 565357096 T^{3} + 283420832722 T^{4} + 565357096 p^{3} T^{5} + 1167226 p^{6} T^{6} + 1456 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1324 T + 2043976 T^{2} + 1611726940 T^{3} + 1469807561518 T^{4} + 1611726940 p^{3} T^{5} + 2043976 p^{6} T^{6} + 1324 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 450 T + 1903832 T^{2} + 631295250 T^{3} + 1545243120894 T^{4} + 631295250 p^{3} T^{5} + 1903832 p^{6} T^{6} + 450 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 3072 T + 5688284 T^{2} + 7201250304 T^{3} + 6916861622502 T^{4} + 7201250304 p^{3} T^{5} + 5688284 p^{6} T^{6} + 3072 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 652 T + 1754332 T^{2} - 1157057716 T^{3} + 2404953539782 T^{4} - 1157057716 p^{3} T^{5} + 1754332 p^{6} T^{6} - 652 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.56054503816477307642447452244, −6.27541295474257218050246558202, −5.87227488915221248872508984693, −5.70466952102789921708955412297, −5.60357201226020354468648835936, −5.43650652225122065767353265206, −5.42177629686587462295808339678, −5.21927231854748104596508944415, −5.11328117649201450051913517960, −4.76458300511517225038525932865, −4.48739823661913007015850299937, −4.36955910148303407806303780605, −4.32882645216477692558027505337, −3.96348087064170576557546987920, −3.57638184949549429285793217821, −3.24534135518849802573170881736, −2.95993286772760325130379001315, −2.87902316241498795960511413372, −2.73513262057433906995558452778, −2.05698249644274196921216776357, −1.74991222174700817016838783528, −1.73281192446457696294183096003, −1.48085365700395338844793837556, −0.904184601321824200233560170188, −0.896430905870101369595148155590, 0, 0, 0, 0, 0.896430905870101369595148155590, 0.904184601321824200233560170188, 1.48085365700395338844793837556, 1.73281192446457696294183096003, 1.74991222174700817016838783528, 2.05698249644274196921216776357, 2.73513262057433906995558452778, 2.87902316241498795960511413372, 2.95993286772760325130379001315, 3.24534135518849802573170881736, 3.57638184949549429285793217821, 3.96348087064170576557546987920, 4.32882645216477692558027505337, 4.36955910148303407806303780605, 4.48739823661913007015850299937, 4.76458300511517225038525932865, 5.11328117649201450051913517960, 5.21927231854748104596508944415, 5.42177629686587462295808339678, 5.43650652225122065767353265206, 5.60357201226020354468648835936, 5.70466952102789921708955412297, 5.87227488915221248872508984693, 6.27541295474257218050246558202, 6.56054503816477307642447452244

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