Properties

Label 4-2156-1.1-c1e2-0-1
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4-s + 2·5-s − 2·9-s − 11-s − 2·12-s − 2·13-s − 4·15-s + 16-s + 8·17-s + 2·19-s + 2·20-s − 8·23-s − 6·25-s + 10·27-s − 8·29-s − 12·31-s + 2·33-s − 2·36-s + 4·39-s + 16·41-s + 12·43-s − 44-s − 4·45-s − 4·47-s − 2·48-s + 49-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.894·5-s − 2/3·9-s − 0.301·11-s − 0.577·12-s − 0.554·13-s − 1.03·15-s + 1/4·16-s + 1.94·17-s + 0.458·19-s + 0.447·20-s − 1.66·23-s − 6/5·25-s + 1.92·27-s − 1.48·29-s − 2.15·31-s + 0.348·33-s − 1/3·36-s + 0.640·39-s + 2.49·41-s + 1.82·43-s − 0.150·44-s − 0.596·45-s − 0.583·47-s − 0.288·48-s + 1/7·49-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

\(L(1)\) \(\approx\) \(0.5579419439\)
\(L(\frac12)\) \(\approx\) \(0.5579419439\)
\(L(\frac{3}{2})\) 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_1$ \( ( 1 - T )( 1 + T ) \)
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$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−18.3499537320, −18.2120541317, −17.5288971196, −17.2128532162, −16.5547739789, −16.3370810014, −15.7955315522, −14.6077730657, −14.5207401195, −13.9541620063, −13.1027441346, −12.3052609901, −12.1245590012, −11.2313614144, −11.0489021393, −10.0445481053, −9.76554711946, −8.91952040154, −7.58187479426, −7.57571100089, −6.02643591301, −5.60617960823, −5.57928681743, −3.78476385236, −2.30984363522, 2.30984363522, 3.78476385236, 5.57928681743, 5.60617960823, 6.02643591301, 7.57571100089, 7.58187479426, 8.91952040154, 9.76554711946, 10.0445481053, 11.0489021393, 11.2313614144, 12.1245590012, 12.3052609901, 13.1027441346, 13.9541620063, 14.5207401195, 14.6077730657, 15.7955315522, 16.3370810014, 16.5547739789, 17.2128532162, 17.5288971196, 18.2120541317, 18.3499537320

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