Properties

Label 4-816-1.1-c1e2-0-0
Degree $4$
Conductor $816$
Sign $1$
Analytic cond. $0.0520288$
Root an. cond. $0.477596$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s − 2·11-s − 12-s + 2·14-s + 16-s − 5·17-s + 4·19-s + 2·21-s + 2·22-s − 2·23-s + 24-s + 2·25-s + 4·27-s − 2·28-s + 8·29-s + 2·31-s − 32-s + 2·33-s + 5·34-s − 8·37-s − 4·38-s + 4·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 0.603·11-s − 0.288·12-s + 0.534·14-s + 1/4·16-s − 1.21·17-s + 0.917·19-s + 0.436·21-s + 0.426·22-s − 0.417·23-s + 0.204·24-s + 2/5·25-s + 0.769·27-s − 0.377·28-s + 1.48·29-s + 0.359·31-s − 0.176·32-s + 0.348·33-s + 0.857·34-s − 1.31·37-s − 0.648·38-s + 0.624·41-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 816,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3078707906\)
\(L(\frac12)\) \(\approx\) \(0.3078707906\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$D_{4}$ \( 1 + 18 T + 166 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$D_{4}$ \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−19.7456619493, −19.2340384236, −18.5811365388, −17.9933882063, −17.6203558362, −17.0792170868, −16.2322617871, −15.8316837652, −15.6362383357, −14.5486772983, −13.8355168830, −13.1753817804, −12.3989857266, −11.9099032971, −11.0931410729, −10.4538652996, −9.93956799794, −9.02514647032, −8.40788928395, −7.34350018754, −6.63216873792, −5.82717930465, −4.65562154609, −2.92100868367, 2.92100868367, 4.65562154609, 5.82717930465, 6.63216873792, 7.34350018754, 8.40788928395, 9.02514647032, 9.93956799794, 10.4538652996, 11.0931410729, 11.9099032971, 12.3989857266, 13.1753817804, 13.8355168830, 14.5486772983, 15.6362383357, 15.8316837652, 16.2322617871, 17.0792170868, 17.6203558362, 17.9933882063, 18.5811365388, 19.2340384236, 19.7456619493

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