Properties

Label 4-686-1.1-c1e2-0-0
Degree $4$
Conductor $686$
Sign $1$
Analytic cond. $0.0437399$
Root an. cond. $0.457319$
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  − 2·3-s − 4-s + 7-s − 2·8-s − 2·9-s + 4·11-s + 2·12-s − 4·13-s + 16-s + 6·17-s + 2·19-s − 2·21-s + 8·23-s + 4·24-s − 10·25-s + 10·27-s − 28-s − 4·29-s − 4·31-s + 4·32-s − 8·33-s + 2·36-s − 4·37-s + 8·39-s + 6·41-s − 4·43-s − 4·44-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 0.377·7-s − 0.707·8-s − 2/3·9-s + 1.20·11-s + 0.577·12-s − 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.436·21-s + 1.66·23-s + 0.816·24-s − 2·25-s + 1.92·27-s − 0.188·28-s − 0.742·29-s − 0.718·31-s + 0.707·32-s − 1.39·33-s + 1/3·36-s − 0.657·37-s + 1.28·39-s + 0.937·41-s − 0.609·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(686\)    =    \(2 \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(0.0437399\)
Root analytic conductor: \(0.457319\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 686,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3192126330\)
\(L(\frac12)\) \(\approx\) \(0.3192126330\)
\(L(\frac{3}{2})\) 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$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
7$C_1$ \( 1 - T \)
good3$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−19.8926107789, −19.5800270198, −18.9333083528, −18.2120541317, −17.5895382325, −17.2128532162, −16.9141484172, −16.3370810014, −15.2564971903, −14.6077730657, −14.3678900325, −13.5582912652, −12.3052609901, −12.2794334476, −11.3085026503, −11.2313614144, −9.76554711946, −9.48952508504, −8.49812018178, −7.57571100089, −6.47803659589, −5.57928681743, −5.08673463815, −3.45773984942, 3.45773984942, 5.08673463815, 5.57928681743, 6.47803659589, 7.57571100089, 8.49812018178, 9.48952508504, 9.76554711946, 11.2313614144, 11.3085026503, 12.2794334476, 12.3052609901, 13.5582912652, 14.3678900325, 14.6077730657, 15.2564971903, 16.3370810014, 16.9141484172, 17.2128532162, 17.5895382325, 18.2120541317, 18.9333083528, 19.5800270198, 19.8926107789

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