Properties

Label 4-2156-1.1-c1e2-0-3
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 3·4-s − 4·5-s + 4·6-s − 4·8-s − 2·9-s + 8·10-s − 11-s − 6·12-s − 2·13-s + 8·15-s + 5·16-s + 2·17-s + 4·18-s − 4·19-s − 12·20-s + 2·22-s + 4·23-s + 8·24-s + 6·25-s + 4·26-s + 10·27-s − 8·29-s − 16·30-s − 6·31-s − 6·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s + 1.63·6-s − 1.41·8-s − 2/3·9-s + 2.52·10-s − 0.301·11-s − 1.73·12-s − 0.554·13-s + 2.06·15-s + 5/4·16-s + 0.485·17-s + 0.942·18-s − 0.917·19-s − 2.68·20-s + 0.426·22-s + 0.834·23-s + 1.63·24-s + 6/5·25-s + 0.784·26-s + 1.92·27-s − 1.48·29-s − 2.92·30-s − 1.07·31-s − 1.06·32-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2156,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−18.9755969718, −18.2120541317, −18.2069927431, −17.2128532162, −16.9307018338, −16.3580045805, −16.3370810014, −15.3001198268, −14.9146375694, −14.6077730657, −13.2488477541, −12.3052609901, −12.2206319828, −11.2313614144, −11.1914967506, −10.8242545693, −9.76554711946, −9.02559149679, −8.38389482609, −7.57571100089, −7.41620352139, −6.22872197038, −5.57928681743, −4.35107164129, −2.98342596761, 0, 2.98342596761, 4.35107164129, 5.57928681743, 6.22872197038, 7.41620352139, 7.57571100089, 8.38389482609, 9.02559149679, 9.76554711946, 10.8242545693, 11.1914967506, 11.2313614144, 12.2206319828, 12.3052609901, 13.2488477541, 14.6077730657, 14.9146375694, 15.3001198268, 16.3370810014, 16.3580045805, 16.9307018338, 17.2128532162, 18.2069927431, 18.2120541317, 18.9755969718

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