Properties

Label 4-1109-1.1-c1e2-0-0
Degree $4$
Conductor $1109$
Sign $1$
Analytic cond. $0.0707108$
Root an. cond. $0.515669$
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  − 2·2-s + 3·5-s + 7-s + 4·8-s − 6·10-s − 5·11-s − 13-s − 2·14-s − 4·16-s + 3·17-s − 9·19-s + 10·22-s − 3·23-s + 2·25-s + 2·26-s + 5·29-s − 31-s − 6·34-s + 3·35-s + 18·38-s + 12·40-s + 5·41-s + 6·43-s + 6·46-s − 7·49-s − 4·50-s − 53-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.34·5-s + 0.377·7-s + 1.41·8-s − 1.89·10-s − 1.50·11-s − 0.277·13-s − 0.534·14-s − 16-s + 0.727·17-s − 2.06·19-s + 2.13·22-s − 0.625·23-s + 2/5·25-s + 0.392·26-s + 0.928·29-s − 0.179·31-s − 1.02·34-s + 0.507·35-s + 2.91·38-s + 1.89·40-s + 0.780·41-s + 0.914·43-s + 0.884·46-s − 49-s − 0.565·50-s − 0.137·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1109 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1109\)
Sign: $1$
Analytic conductor: \(0.0707108\)
Root analytic conductor: \(0.515669\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1109} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1109,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2885064565\)
\(L(\frac12)\) \(\approx\) \(0.2885064565\)
\(L(\frac{3}{2})\) 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)$
bad1109$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 50 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 9 T + 53 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$D_{4}$ \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 4 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 61 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 8 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 7 T + 106 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 150 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 5 T + 135 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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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

−19.5082415633, −19.0234623439, −18.6347746586, −17.9982729700, −17.7078996587, −17.3107248098, −16.8557025546, −16.1093797206, −15.4802633775, −14.4158764913, −14.2693216873, −13.3232931067, −13.0112683442, −12.3279847791, −11.0743273627, −10.4507701525, −10.0283749808, −9.58517150303, −8.68182783245, −8.23349910033, −7.56845968617, −6.29572249493, −5.42702336844, −4.48303726399, −2.25941107232, 2.25941107232, 4.48303726399, 5.42702336844, 6.29572249493, 7.56845968617, 8.23349910033, 8.68182783245, 9.58517150303, 10.0283749808, 10.4507701525, 11.0743273627, 12.3279847791, 13.0112683442, 13.3232931067, 14.2693216873, 14.4158764913, 15.4802633775, 16.1093797206, 16.8557025546, 17.3107248098, 17.7078996587, 17.9982729700, 18.6347746586, 19.0234623439, 19.5082415633

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