Properties

Label 8-1792e4-1.1-c3e4-0-1
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 $0$

Origins

Origins of factors

Downloads

Learn more

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: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(4.612804877\)
\(L(\frac12)\) \(\approx\) \(4.612804877\)
\(L(\frac{5}{2})\) 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$C_2 \wr C_2\wr C_2$ \( 1 - 192 T + 3197100 T^{2} - 518194304 T^{3} + 4172546665638 T^{4} - 518194304 p^{3} T^{5} + 3197100 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.42678925123111045839744781714, −5.81770730118823792209960382212, −5.78874488615390138911736700483, −5.74641588553499152978448185380, −5.51729803949494835402234875944, −5.24844455528242366318362507124, −4.83307560397835179015911295749, −4.80175427095701691665028359665, −4.27969491777478197153508710565, −3.88213181764830998779885012308, −3.86159283610387424878072357514, −3.81978956970797143014292453671, −3.48974485349875266942384588891, −3.38644900683686448087483849215, −3.11513369561883381117652777912, −3.03545281018653425015010432194, −2.81903040785615204719242386130, −2.15747981870927512825627649339, −2.03517129900038830552400863632, −1.94350289049325870690326625845, −1.48067011290612511771299881607, −0.906466657829368194829004203528, −0.886400232129827975270809937563, −0.42413395749772455274837422692, −0.30756409850774223763801239453, 0.30756409850774223763801239453, 0.42413395749772455274837422692, 0.886400232129827975270809937563, 0.906466657829368194829004203528, 1.48067011290612511771299881607, 1.94350289049325870690326625845, 2.03517129900038830552400863632, 2.15747981870927512825627649339, 2.81903040785615204719242386130, 3.03545281018653425015010432194, 3.11513369561883381117652777912, 3.38644900683686448087483849215, 3.48974485349875266942384588891, 3.81978956970797143014292453671, 3.86159283610387424878072357514, 3.88213181764830998779885012308, 4.27969491777478197153508710565, 4.80175427095701691665028359665, 4.83307560397835179015911295749, 5.24844455528242366318362507124, 5.51729803949494835402234875944, 5.74641588553499152978448185380, 5.78874488615390138911736700483, 5.81770730118823792209960382212, 6.42678925123111045839744781714

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