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Properties

Label 8-966e4-1.1-c3e4-0-2
Degree $8$
Conductor $870780120336$
Sign $1$
Analytic cond. $1.05529\times 10^{7}$
Root an. cond. $7.54955$
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  − 8·2-s + 12·3-s + 40·4-s + 5·5-s − 96·6-s − 28·7-s − 160·8-s + 90·9-s − 40·10-s + 19·11-s + 480·12-s − 73·13-s + 224·14-s + 60·15-s + 560·16-s − 14·17-s − 720·18-s − 51·19-s + 200·20-s − 336·21-s − 152·22-s − 92·23-s − 1.92e3·24-s − 140·25-s + 584·26-s + 540·27-s − 1.12e3·28-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s + 0.447·5-s − 6.53·6-s − 1.51·7-s − 7.07·8-s + 10/3·9-s − 1.26·10-s + 0.520·11-s + 11.5·12-s − 1.55·13-s + 4.27·14-s + 1.03·15-s + 35/4·16-s − 0.199·17-s − 9.42·18-s − 0.615·19-s + 2.23·20-s − 3.49·21-s − 1.47·22-s − 0.834·23-s − 16.3·24-s − 1.11·25-s + 4.40·26-s + 3.84·27-s − 7.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1.05529\times 10^{7}\)
Root analytic conductor: \(7.54955\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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$C_1$ \( ( 1 + p T )^{4} \)
3$C_1$ \( ( 1 - p T )^{4} \)
7$C_1$ \( ( 1 + p T )^{4} \)
23$C_1$ \( ( 1 + p T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - p T + 33 p T^{2} - 241 T^{3} + 15152 T^{4} - 241 p^{3} T^{5} + 33 p^{7} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 19 T + 3035 T^{2} - 11832 T^{3} + 4372752 T^{4} - 11832 p^{3} T^{5} + 3035 p^{6} T^{6} - 19 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 73 T + 9479 T^{2} + 440315 T^{3} + 31340040 T^{4} + 440315 p^{3} T^{5} + 9479 p^{6} T^{6} + 73 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 14 T - 2285 T^{2} + 73560 T^{3} + 46228172 T^{4} + 73560 p^{3} T^{5} - 2285 p^{6} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 51 T + 24311 T^{2} + 882252 T^{3} + 238965536 T^{4} + 882252 p^{3} T^{5} + 24311 p^{6} T^{6} + 51 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 8 p T + 52243 T^{2} + 8346536 T^{3} + 1042367840 T^{4} + 8346536 p^{3} T^{5} + 52243 p^{6} T^{6} + 8 p^{10} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 80 T + 76361 T^{2} - 689794 T^{3} + 2548460844 T^{4} - 689794 p^{3} T^{5} + 76361 p^{6} T^{6} - 80 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 150 T + 169897 T^{2} + 21055806 T^{3} + 12330989340 T^{4} + 21055806 p^{3} T^{5} + 169897 p^{6} T^{6} + 150 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 491 T + 294901 T^{2} + 93211618 T^{3} + 31003035986 T^{4} + 93211618 p^{3} T^{5} + 294901 p^{6} T^{6} + 491 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 481 T + 353325 T^{2} + 108715209 T^{3} + 43315991068 T^{4} + 108715209 p^{3} T^{5} + 353325 p^{6} T^{6} + 481 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 289 T + 60150 T^{2} + 4965553 T^{3} + 6499889074 T^{4} + 4965553 p^{3} T^{5} + 60150 p^{6} T^{6} + 289 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 176 T + 330806 T^{2} + 52716891 T^{3} + 71511305700 T^{4} + 52716891 p^{3} T^{5} + 330806 p^{6} T^{6} + 176 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 212 T + 483330 T^{2} - 5216081 T^{3} + 102766359382 T^{4} - 5216081 p^{3} T^{5} + 483330 p^{6} T^{6} - 212 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 2066 T + 2330398 T^{2} + 1755460825 T^{3} + 968479611452 T^{4} + 1755460825 p^{3} T^{5} + 2330398 p^{6} T^{6} + 2066 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 369 T + 980275 T^{2} + 243163661 T^{3} + 400953834864 T^{4} + 243163661 p^{3} T^{5} + 980275 p^{6} T^{6} + 369 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 1565 T + 1904471 T^{2} + 1519113713 T^{3} + 1040345291904 T^{4} + 1519113713 p^{3} T^{5} + 1904471 p^{6} T^{6} + 1565 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 482 T + 631919 T^{2} + 74134548 T^{3} + 157172222324 T^{4} + 74134548 p^{3} T^{5} + 631919 p^{6} T^{6} + 482 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 2146 T + 3330515 T^{2} + 3414480964 T^{3} + 2795621216600 T^{4} + 3414480964 p^{3} T^{5} + 3330515 p^{6} T^{6} + 2146 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1040 T + 1094529 T^{2} - 858628486 T^{3} + 858407292548 T^{4} - 858628486 p^{3} T^{5} + 1094529 p^{6} T^{6} - 1040 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 1963 T + 2093713 T^{2} - 1189085469 T^{3} + 768350349452 T^{4} - 1189085469 p^{3} T^{5} + 2093713 p^{6} T^{6} - 1963 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 818 T + 1998497 T^{2} + 235499822 T^{3} + 1457132033180 T^{4} + 235499822 p^{3} T^{5} + 1998497 p^{6} T^{6} + 818 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.50346785798058726818174487411, −7.22830744330518769532439727660, −6.90338162818058269489740405762, −6.68827940673000118238749756099, −6.67756507791107982977463596390, −6.20583204534692070214944982954, −6.02017082473379804187656883443, −5.97151636275458924360095425325, −5.76601793833018279984618479662, −5.02272860474241925319245055921, −4.93403601082495613113370501485, −4.57334580757735090486056625340, −4.48119530247518451967125360836, −3.68275665359415063708540783624, −3.65645952573521579604031897377, −3.52200573933085202407773983761, −3.36357829293459292098709003644, −2.80507694017243662967691076203, −2.61770339099616940674614498522, −2.54646127836588584059394727671, −2.18857028197590407903717124016, −1.85010482182305820190737800365, −1.51631098170775681564934765798, −1.45178833613447801449564461512, −1.22266543281403195085415495765, 0, 0, 0, 0, 1.22266543281403195085415495765, 1.45178833613447801449564461512, 1.51631098170775681564934765798, 1.85010482182305820190737800365, 2.18857028197590407903717124016, 2.54646127836588584059394727671, 2.61770339099616940674614498522, 2.80507694017243662967691076203, 3.36357829293459292098709003644, 3.52200573933085202407773983761, 3.65645952573521579604031897377, 3.68275665359415063708540783624, 4.48119530247518451967125360836, 4.57334580757735090486056625340, 4.93403601082495613113370501485, 5.02272860474241925319245055921, 5.76601793833018279984618479662, 5.97151636275458924360095425325, 6.02017082473379804187656883443, 6.20583204534692070214944982954, 6.67756507791107982977463596390, 6.68827940673000118238749756099, 6.90338162818058269489740405762, 7.22830744330518769532439727660, 7.50346785798058726818174487411

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