Properties

Label 8-6762e4-1.1-c1e4-0-0
Degree $8$
Conductor $2.091\times 10^{15}$
Sign $1$
Analytic cond. $8.49980\times 10^{6}$
Root an. cond. $7.34811$
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 + 2·5-s + 16·6-s − 20·8-s + 10·9-s − 8·10-s + 6·11-s − 40·12-s − 2·13-s − 8·15-s + 35·16-s − 2·17-s − 40·18-s − 6·19-s + 20·20-s − 24·22-s + 4·23-s + 80·24-s − 5·25-s + 8·26-s − 20·27-s − 10·29-s + 32·30-s − 14·31-s − 56·32-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s + 0.894·5-s + 6.53·6-s − 7.07·8-s + 10/3·9-s − 2.52·10-s + 1.80·11-s − 11.5·12-s − 0.554·13-s − 2.06·15-s + 35/4·16-s − 0.485·17-s − 9.42·18-s − 1.37·19-s + 4.47·20-s − 5.11·22-s + 0.834·23-s + 16.3·24-s − 25-s + 1.56·26-s − 3.84·27-s − 1.85·29-s + 5.84·30-s − 2.51·31-s − 9.89·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(8.49980\times 10^{6}\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6762} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1820782346\)
\(L(\frac12)\) \(\approx\) \(0.1820782346\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
7 \( 1 \)
23$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 9 T^{2} - 18 T^{3} + 48 T^{4} - 18 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 2 p T^{2} - 28 T^{3} + 43 T^{4} - 28 p T^{5} + 2 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 18 T^{2} - 34 T^{3} + 74 T^{4} - 34 p T^{5} + 18 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 43 T^{2} + 20 T^{3} + 822 T^{4} + 20 p T^{5} + 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 42 T^{2} + 262 T^{3} + 1170 T^{4} + 262 p T^{5} + 42 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 106 T^{2} + 744 T^{3} + 4655 T^{4} + 744 p T^{5} + 106 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 129 T^{2} + 882 T^{3} + 5440 T^{4} + 882 p T^{5} + 129 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 126 T^{2} - 650 T^{3} + 6618 T^{4} - 650 p T^{5} + 126 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 154 T^{2} - 1006 T^{3} + 9210 T^{4} - 1006 p T^{5} + 154 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 109 T^{2} + 358 T^{3} + 3240 T^{4} + 358 p T^{5} + 109 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 213 T^{2} - 930 T^{3} + 16948 T^{4} - 930 p T^{5} + 213 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 333 T^{2} - 3096 T^{3} + 25228 T^{4} - 3096 p T^{5} + 333 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 28 T + 8 p T^{2} + 5796 T^{3} + 52222 T^{4} + 5796 p T^{5} + 8 p^{3} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 28 T + 420 T^{2} - 4396 T^{3} + 38294 T^{4} - 4396 p T^{5} + 420 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 351 T^{2} - 3474 T^{3} + 39778 T^{4} - 3474 p T^{5} + 351 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 189 T^{2} - 446 T^{3} + 15312 T^{4} - 446 p T^{5} + 189 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 90 T^{2} - 296 T^{3} + 2951 T^{4} - 296 p T^{5} + 90 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 289 T^{2} - 2310 T^{3} + 30488 T^{4} - 2310 p T^{5} + 289 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 270 T^{2} + 2394 T^{3} + 29146 T^{4} + 2394 p T^{5} + 270 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 69 T^{2} + 112 T^{3} + 18456 T^{4} + 112 p T^{5} + 69 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.82736134209048122582018324216, −5.50960626237394540237307161896, −5.34670144703010392391073988632, −5.24957713901389135412716623286, −5.18297054275459945096255831533, −4.54568732891428114610549170811, −4.53250047663261252834368693485, −4.35315445871337384944617272902, −4.15295736986589486305135903156, −3.78019718096811064255586789381, −3.56589061479397468092632003167, −3.53038521178525238073835142333, −3.51241069617902071238114461467, −2.78948479977735512286874614245, −2.46691615361862362854079444892, −2.43877879036850331666999715498, −2.14364992413274269352002652176, −1.91784787055961230336900353961, −1.86190614440183773283396894417, −1.47013736584370937487735917441, −1.43748968176287777293829702175, −0.898983693318008273191383283157, −0.77006335000784245261392475130, −0.53366150296210084484886168875, −0.15754336048952132705499490300, 0.15754336048952132705499490300, 0.53366150296210084484886168875, 0.77006335000784245261392475130, 0.898983693318008273191383283157, 1.43748968176287777293829702175, 1.47013736584370937487735917441, 1.86190614440183773283396894417, 1.91784787055961230336900353961, 2.14364992413274269352002652176, 2.43877879036850331666999715498, 2.46691615361862362854079444892, 2.78948479977735512286874614245, 3.51241069617902071238114461467, 3.53038521178525238073835142333, 3.56589061479397468092632003167, 3.78019718096811064255586789381, 4.15295736986589486305135903156, 4.35315445871337384944617272902, 4.53250047663261252834368693485, 4.54568732891428114610549170811, 5.18297054275459945096255831533, 5.24957713901389135412716623286, 5.34670144703010392391073988632, 5.50960626237394540237307161896, 5.82736134209048122582018324216

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