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Properties

Label 8-7350e4-1.1-c1e4-0-1
Degree $8$
Conductor $2.918\times 10^{15}$
Sign $1$
Analytic cond. $1.18647\times 10^{7}$
Root an. cond. $7.66094$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·3-s + 10·4-s + 16·6-s + 20·8-s + 10·9-s + 40·12-s + 35·16-s + 4·17-s + 40·18-s + 12·19-s + 4·23-s + 80·24-s + 20·27-s − 8·29-s + 16·31-s + 56·32-s + 16·34-s + 100·36-s − 8·37-s + 48·38-s + 12·41-s + 4·43-s + 16·46-s + 12·47-s + 140·48-s + 16·51-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 6.53·6-s + 7.07·8-s + 10/3·9-s + 11.5·12-s + 35/4·16-s + 0.970·17-s + 9.42·18-s + 2.75·19-s + 0.834·23-s + 16.3·24-s + 3.84·27-s − 1.48·29-s + 2.87·31-s + 9.89·32-s + 2.74·34-s + 50/3·36-s − 1.31·37-s + 7.78·38-s + 1.87·41-s + 0.609·43-s + 2.35·46-s + 1.75·47-s + 20.2·48-s + 2.24·51-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.18647\times 10^{7}\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(388.1673539\)
\(L(\frac12)\) \(\approx\) \(388.1673539\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
3$C_1$ \( ( 1 - T )^{4} \)
5 \( 1 \)
7 \( 1 \)
good11$C_2 \wr C_2\wr C_2$ \( 1 + 26 T^{2} - 16 T^{3} + 362 T^{4} - 16 p T^{5} + 26 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T^{2} + 48 T^{3} + 322 T^{4} + 48 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 36 T^{2} - 196 T^{3} + 770 T^{4} - 196 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 88 T^{2} - 460 T^{3} + 2174 T^{4} - 460 p T^{5} + 88 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 56 T^{2} - 12 p T^{3} + 1646 T^{4} - 12 p^{2} T^{5} + 56 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 50 T^{2} + 360 T^{3} + 2786 T^{4} + 360 p T^{5} + 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 202 T^{2} - 1616 T^{3} + 10666 T^{4} - 1616 p T^{5} + 202 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 106 T^{2} + 768 T^{3} + 5306 T^{4} + 768 p T^{5} + 106 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 190 T^{2} - 1356 T^{3} + 11842 T^{4} - 1356 p T^{5} + 190 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 62 T^{2} - 148 T^{3} + 2370 T^{4} - 148 p T^{5} + 62 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 158 T^{2} - 1180 T^{3} + 9410 T^{4} - 1180 p T^{5} + 158 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 320 T^{2} - 3340 T^{3} + 28366 T^{4} - 3340 p T^{5} + 320 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 288 T^{2} - 2804 T^{3} + 24014 T^{4} - 2804 p T^{5} + 288 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 118 T^{2} - 684 T^{3} + 6306 T^{4} - 684 p T^{5} + 118 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 254 T^{2} + 788 T^{3} + 25090 T^{4} + 788 p T^{5} + 254 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 248 T^{2} + 2788 T^{3} + 28078 T^{4} + 2788 p T^{5} + 248 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 206 T^{2} + 2092 T^{3} + 19170 T^{4} + 2092 p T^{5} + 206 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 140 T^{2} + 104 T^{3} + 6054 T^{4} + 104 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 244 T^{2} - 2056 T^{3} + 26854 T^{4} - 2056 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 44 T + 1014 T^{2} - 15436 T^{3} + 169698 T^{4} - 15436 p T^{5} + 1014 p^{2} T^{6} - 44 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 374 T^{2} - 3636 T^{3} + 43362 T^{4} - 3636 p T^{5} + 374 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.63289756845794121258831751440, −5.17605805937075357452710477725, −5.11697373041346124417376959914, −5.00610673377613956261998960819, −4.88114229137090114631395048623, −4.37782425601690433108604542064, −4.31825223218995147814513416166, −4.23184382733025515340356002022, −4.13244173664602744956671485391, −3.64471720503782317464885524021, −3.53696078302411505666525004830, −3.50326126787412380005162579643, −3.48006963483029210604499674330, −2.94160417204587660326820643741, −2.88851263826486144639511348755, −2.78627024810140086196868936560, −2.74934302333561794743622030683, −2.14128889656692343351598505722, −2.06543386989620487412218596979, −1.99992172742794129851399699304, −1.93344072929788154195720176126, −1.01830015939411037786495103992, −0.975955143813722509429284577011, −0.942088417644440448981673667906, −0.899711084138143284354518122232, 0.899711084138143284354518122232, 0.942088417644440448981673667906, 0.975955143813722509429284577011, 1.01830015939411037786495103992, 1.93344072929788154195720176126, 1.99992172742794129851399699304, 2.06543386989620487412218596979, 2.14128889656692343351598505722, 2.74934302333561794743622030683, 2.78627024810140086196868936560, 2.88851263826486144639511348755, 2.94160417204587660326820643741, 3.48006963483029210604499674330, 3.50326126787412380005162579643, 3.53696078302411505666525004830, 3.64471720503782317464885524021, 4.13244173664602744956671485391, 4.23184382733025515340356002022, 4.31825223218995147814513416166, 4.37782425601690433108604542064, 4.88114229137090114631395048623, 5.00610673377613956261998960819, 5.11697373041346124417376959914, 5.17605805937075357452710477725, 5.63289756845794121258831751440

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