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Properties

Label 8-1792e4-1.1-c3e4-0-4
Degree $8$
Conductor $1.031\times 10^{13}$
Sign $1$
Analytic cond. $1.24973\times 10^{8}$
Root an. cond. $10.2825$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 14·5-s + 28·7-s − 32·9-s − 56·11-s + 46·13-s + 84·15-s + 136·17-s − 58·19-s − 168·21-s + 36·23-s − 120·25-s + 298·27-s − 244·29-s − 80·31-s + 336·33-s − 392·35-s + 116·37-s − 276·39-s − 376·41-s + 608·43-s + 448·45-s − 504·47-s + 490·49-s − 816·51-s − 88·53-s + 784·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.25·5-s + 1.51·7-s − 1.18·9-s − 1.53·11-s + 0.981·13-s + 1.44·15-s + 1.94·17-s − 0.700·19-s − 1.74·21-s + 0.326·23-s − 0.959·25-s + 2.12·27-s − 1.56·29-s − 0.463·31-s + 1.77·33-s − 1.89·35-s + 0.515·37-s − 1.13·39-s − 1.43·41-s + 2.15·43-s + 1.48·45-s − 1.56·47-s + 10/7·49-s − 2.24·51-s − 0.228·53-s + 1.92·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.24973\times 10^{8}\)
Root analytic conductor: \(10.2825\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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)$
bad2 \( 1 \)
7$C_1$ \( ( 1 - p T )^{4} \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T + 68 T^{2} + 302 T^{3} + 2102 T^{4} + 302 p^{3} T^{5} + 68 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 316 T^{2} + 18 p^{3} T^{3} + 39446 T^{4} + 18 p^{6} T^{5} + 316 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 56 T + 2780 T^{2} + 130424 T^{3} + 5262166 T^{4} + 130424 p^{3} T^{5} + 2780 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 46 T + 5180 T^{2} - 228570 T^{3} + 16614646 T^{4} - 228570 p^{3} T^{5} + 5180 p^{6} T^{6} - 46 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 8 p T + 22076 T^{2} - 109112 p T^{3} + 168489670 T^{4} - 109112 p^{4} T^{5} + 22076 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 58 T + 21284 T^{2} + 1222770 T^{3} + 200115766 T^{4} + 1222770 p^{3} T^{5} + 21284 p^{6} T^{6} + 58 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 36 T + 11068 T^{2} + 2483404 T^{3} - 109072922 T^{4} + 2483404 p^{3} T^{5} + 11068 p^{6} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 244 T + 3380 p T^{2} + 16054732 T^{3} + 3637039542 T^{4} + 16054732 p^{3} T^{5} + 3380 p^{7} T^{6} + 244 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 80 T + 98108 T^{2} + 6698256 T^{3} + 4121040902 T^{4} + 6698256 p^{3} T^{5} + 98108 p^{6} T^{6} + 80 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 116 T + 112292 T^{2} - 457596 p T^{3} + 191806558 p T^{4} - 457596 p^{4} T^{5} + 112292 p^{6} T^{6} - 116 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 376 T + 212316 T^{2} + 51001224 T^{3} + 18244997158 T^{4} + 51001224 p^{3} T^{5} + 212316 p^{6} T^{6} + 376 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 608 T + 332604 T^{2} - 113978720 T^{3} + 38738414102 T^{4} - 113978720 p^{3} T^{5} + 332604 p^{6} T^{6} - 608 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 504 T + 371452 T^{2} + 143870104 T^{3} + 56727343558 T^{4} + 143870104 p^{3} T^{5} + 371452 p^{6} T^{6} + 504 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 88 T + 35956 T^{2} - 755896 p T^{3} + 10958668662 T^{4} - 755896 p^{4} T^{5} + 35956 p^{6} T^{6} + 88 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 38 T + 123316 T^{2} - 6020190 T^{3} + 81607069814 T^{4} - 6020190 p^{3} T^{5} + 123316 p^{6} T^{6} - 38 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 130 T + 528540 T^{2} + 57273642 T^{3} + 122928435190 T^{4} + 57273642 p^{3} T^{5} + 528540 p^{6} T^{6} - 130 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 1668 T + 1986284 T^{2} + 1562559876 T^{3} + 1003433820182 T^{4} + 1562559876 p^{3} T^{5} + 1986284 p^{6} T^{6} + 1668 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 408 T + 545308 T^{2} + 167809544 T^{3} + 48366937318 T^{4} + 167809544 p^{3} T^{5} + 545308 p^{6} T^{6} - 408 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 576 T + 1219532 T^{2} - 601885056 T^{3} + 666382613702 T^{4} - 601885056 p^{3} T^{5} + 1219532 p^{6} T^{6} - 576 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 472 T + 994172 T^{2} + 344402424 T^{3} + 659437576006 T^{4} + 344402424 p^{3} T^{5} + 994172 p^{6} T^{6} + 472 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 2058 T + 1798900 T^{2} + 411223106 T^{3} - 157636635434 T^{4} + 411223106 p^{3} T^{5} + 1798900 p^{6} T^{6} + 2058 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 344 T + 881148 T^{2} + 629036712 T^{3} + 820148016934 T^{4} + 629036712 p^{3} T^{5} + 881148 p^{6} T^{6} + 344 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 192 T + 3197100 T^{2} - 518194304 T^{3} + 4172546665638 T^{4} - 518194304 p^{3} T^{5} + 3197100 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.52413181972370908709099462003, −6.18056437008900219557106789528, −6.09014913470435729110598913490, −6.02686050843216555079967256613, −5.76532525652297377302965062692, −5.58326543137823644574954481850, −5.36311930049320505030646971581, −5.25230538150712644261146469165, −5.07593449143211715471819244677, −4.70135715713273454959417447124, −4.52139366061277331973485692261, −4.42045268705660183099072711029, −4.10117627295523521227146696040, −3.63036867502243540476636639416, −3.44357652649873109216329791939, −3.43891174637736315802820963399, −3.39992331184306442426356553067, −2.70557477932443721004705904276, −2.46613358224754919385070759031, −2.33604667100942381991229605091, −2.16196725781775587353825767257, −1.43316141228069867248446465232, −1.35653399137530402094022511506, −1.20914743030456768017850829589, −0.819064322595711263927052512238, 0, 0, 0, 0, 0.819064322595711263927052512238, 1.20914743030456768017850829589, 1.35653399137530402094022511506, 1.43316141228069867248446465232, 2.16196725781775587353825767257, 2.33604667100942381991229605091, 2.46613358224754919385070759031, 2.70557477932443721004705904276, 3.39992331184306442426356553067, 3.43891174637736315802820963399, 3.44357652649873109216329791939, 3.63036867502243540476636639416, 4.10117627295523521227146696040, 4.42045268705660183099072711029, 4.52139366061277331973485692261, 4.70135715713273454959417447124, 5.07593449143211715471819244677, 5.25230538150712644261146469165, 5.36311930049320505030646971581, 5.58326543137823644574954481850, 5.76532525652297377302965062692, 6.02686050843216555079967256613, 6.09014913470435729110598913490, 6.18056437008900219557106789528, 6.52413181972370908709099462003

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