Properties

Label 8-28e8-1.1-c1e4-0-6
Degree $8$
Conductor $377801998336$
Sign $1$
Analytic cond. $1535.93$
Root an. cond. $2.50205$
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  + 2·2-s − 2·3-s + 2·4-s − 2·5-s − 4·6-s + 4·8-s + 2·9-s − 4·10-s − 2·11-s − 4·12-s + 4·13-s + 4·15-s + 8·16-s − 4·17-s + 4·18-s + 6·19-s − 4·20-s − 4·22-s − 8·24-s + 2·25-s + 8·26-s + 8·27-s + 12·29-s + 8·30-s − 16·31-s + 8·32-s + 4·33-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s − 1.63·6-s + 1.41·8-s + 2/3·9-s − 1.26·10-s − 0.603·11-s − 1.15·12-s + 1.10·13-s + 1.03·15-s + 2·16-s − 0.970·17-s + 0.942·18-s + 1.37·19-s − 0.894·20-s − 0.852·22-s − 1.63·24-s + 2/5·25-s + 1.56·26-s + 1.53·27-s + 2.22·29-s + 1.46·30-s − 2.87·31-s + 1.41·32-s + 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1535.93\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{784} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.473278579\)
\(L(\frac12)\) \(\approx\) \(4.473278579\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good3$C_2^3$ \( 1 + 2 T + 2 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
11$C_2^3$ \( 1 + 2 T + 2 T^{2} - 40 T^{3} - 161 T^{4} - 40 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^3$ \( 1 - 6 T + 18 T^{2} + 120 T^{3} - 721 T^{4} + 120 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^3$ \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^3$ \( 1 + 6 T + 18 T^{2} - 336 T^{3} - 2377 T^{4} - 336 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
59$C_2^3$ \( 1 + 6 T + 18 T^{2} - 600 T^{3} - 5281 T^{4} - 600 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^3$ \( 1 + 18 T + 162 T^{2} + 720 T^{3} + 2759 T^{4} + 720 p T^{5} + 162 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^3$ \( 1 - 10 T + 50 T^{2} + 840 T^{3} - 8689 T^{4} + 840 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 162 T^{2} + 18323 T^{4} + 162 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
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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

−7.11386553298184052910643055818, −7.09266228288039239077231175608, −7.01672793883560665313982491579, −6.89127760919081632955773886059, −6.24045477026544824353755734131, −6.05211541274121437411088437142, −6.01416840664914436388468621447, −5.60939877429752001293440665617, −5.56925844996236010240781353349, −5.28190650925256759172563825635, −5.06237604054954969541819905666, −4.51609685287121078803663308180, −4.47329661958484746521338452563, −4.45738847966052316452135854193, −4.30296248885310855361922526969, −3.59611010321157134259589652330, −3.54977982277054317555619917832, −3.27753167638859460829500718492, −3.14594347745426050238038872269, −2.42153094949907804958072285165, −2.34333829899732633025400223447, −1.94805754406740269319251515750, −1.14414461070119456571443506797, −1.07587071498164173990890850287, −0.56495862658599632052662303324, 0.56495862658599632052662303324, 1.07587071498164173990890850287, 1.14414461070119456571443506797, 1.94805754406740269319251515750, 2.34333829899732633025400223447, 2.42153094949907804958072285165, 3.14594347745426050238038872269, 3.27753167638859460829500718492, 3.54977982277054317555619917832, 3.59611010321157134259589652330, 4.30296248885310855361922526969, 4.45738847966052316452135854193, 4.47329661958484746521338452563, 4.51609685287121078803663308180, 5.06237604054954969541819905666, 5.28190650925256759172563825635, 5.56925844996236010240781353349, 5.60939877429752001293440665617, 6.01416840664914436388468621447, 6.05211541274121437411088437142, 6.24045477026544824353755734131, 6.89127760919081632955773886059, 7.01672793883560665313982491579, 7.09266228288039239077231175608, 7.11386553298184052910643055818

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