Properties

Label 4-816-1.1-c1e2-0-1
Degree $4$
Conductor $816$
Sign $1$
Analytic cond. $0.0520288$
Root an. cond. $0.477596$
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-s − 3·3-s + 4-s − 2·5-s − 3·6-s − 4·7-s + 8-s + 4·9-s − 2·10-s + 10·11-s − 3·12-s − 4·14-s + 6·15-s + 16-s + 17-s + 4·18-s − 8·19-s − 2·20-s + 12·21-s + 10·22-s − 8·23-s − 3·24-s − 6·25-s − 4·28-s + 6·29-s + 6·30-s + 4·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s − 1.22·6-s − 1.51·7-s + 0.353·8-s + 4/3·9-s − 0.632·10-s + 3.01·11-s − 0.866·12-s − 1.06·14-s + 1.54·15-s + 1/4·16-s + 0.242·17-s + 0.942·18-s − 1.83·19-s − 0.447·20-s + 2.61·21-s + 2.13·22-s − 1.66·23-s − 0.612·24-s − 6/5·25-s − 0.755·28-s + 1.11·29-s + 1.09·30-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.0520288\)
Root analytic conductor: \(0.477596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{816} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 816,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4039570128\)
\(L(\frac12)\) \(\approx\) \(0.4039570128\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.9292301410, −19.8934316645, −19.1551537949, −19.1494492613, −17.6707519500, −17.4469801976, −16.6317948443, −16.6272610009, −15.7312805996, −15.4732516475, −14.2197849424, −14.1819882616, −12.8595646329, −12.3172568607, −11.9583441737, −11.5820474614, −10.8489466872, −9.96467191573, −9.28607198826, −8.09899069409, −6.65330731261, −6.42897107073, −5.99911855172, −4.25303028693, −3.90229547123, 3.90229547123, 4.25303028693, 5.99911855172, 6.42897107073, 6.65330731261, 8.09899069409, 9.28607198826, 9.96467191573, 10.8489466872, 11.5820474614, 11.9583441737, 12.3172568607, 12.8595646329, 14.1819882616, 14.2197849424, 15.4732516475, 15.7312805996, 16.6272610009, 16.6317948443, 17.4469801976, 17.6707519500, 19.1494492613, 19.1551537949, 19.8934316645, 19.9292301410

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