Properties

Label 4-2156-1.1-c1e2-0-0
Degree $4$
Conductor $2156$
Sign $1$
Analytic cond. $0.137468$
Root an. cond. $0.608906$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 2·9-s − 4·10-s + 11-s − 8·13-s + 5·16-s + 6·17-s + 4·18-s + 6·19-s + 6·20-s − 2·22-s + 4·23-s − 6·25-s + 16·26-s − 4·29-s − 14·31-s − 6·32-s − 12·34-s − 6·36-s − 4·37-s − 12·38-s − 8·40-s + 6·41-s + 4·43-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.41·8-s − 2/3·9-s − 1.26·10-s + 0.301·11-s − 2.21·13-s + 5/4·16-s + 1.45·17-s + 0.942·18-s + 1.37·19-s + 1.34·20-s − 0.426·22-s + 0.834·23-s − 6/5·25-s + 3.13·26-s − 0.742·29-s − 2.51·31-s − 1.06·32-s − 2.05·34-s − 36-s − 0.657·37-s − 1.94·38-s − 1.26·40-s + 0.937·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.137468\)
Root analytic conductor: \(0.608906\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2156} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2156,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4028676753\)
\(L(\frac12)\) \(\approx\) \(0.4028676753\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−18.5949509848, −18.2120541317, −17.5270697575, −17.2128532162, −16.9003432360, −16.3370810014, −15.7576496326, −14.7279836778, −14.6077730657, −14.0732702299, −13.1223749446, −12.3052609901, −11.9889055567, −11.2313614144, −10.5971046162, −9.76554711946, −9.42342370074, −9.24085361043, −7.87359639769, −7.57571100089, −6.89070428270, −5.57928681743, −5.41745538951, −3.29966281651, −2.06919705445, 2.06919705445, 3.29966281651, 5.41745538951, 5.57928681743, 6.89070428270, 7.57571100089, 7.87359639769, 9.24085361043, 9.42342370074, 9.76554711946, 10.5971046162, 11.2313614144, 11.9889055567, 12.3052609901, 13.1223749446, 14.0732702299, 14.6077730657, 14.7279836778, 15.7576496326, 16.3370810014, 16.9003432360, 17.2128532162, 17.5270697575, 18.2120541317, 18.5949509848

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