Properties

Label 4-1309-1.1-c1e2-0-0
Degree $4$
Conductor $1309$
Sign $1$
Analytic cond. $0.0834629$
Root an. cond. $0.537493$
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-s + 3-s + 4-s − 3·5-s − 6-s + 7-s − 3·8-s − 9-s + 3·10-s + 3·11-s + 12-s − 2·13-s − 14-s − 3·15-s + 16-s − 5·17-s + 18-s + 2·19-s − 3·20-s + 21-s − 3·22-s + 7·23-s − 3·24-s + 5·25-s + 2·26-s + 28-s − 4·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s − 1/3·9-s + 0.948·10-s + 0.904·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s + 0.458·19-s − 0.670·20-s + 0.218·21-s − 0.639·22-s + 1.45·23-s − 0.612·24-s + 25-s + 0.392·26-s + 0.188·28-s − 0.742·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.0834629\)
Root analytic conductor: \(0.537493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1309,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4233005095\)
\(L(\frac12)\) \(\approx\) \(0.4233005095\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$D_{4}$ \( 1 - 5 T + 52 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$D_{4}$ \( 1 - 11 T + 74 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 9 T + 90 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$D_{4}$ \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 19 T + 216 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
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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

−19.5706470116, −19.1297294501, −18.5412106719, −17.9582356663, −17.3563779162, −16.8782224727, −16.1785116913, −15.5315873680, −15.1376917508, −14.6679752241, −14.1312186687, −13.1371488923, −12.5072882973, −11.6603604664, −11.3928378246, −10.9444831981, −9.60578416477, −9.22211166284, −8.46983732551, −8.06812083543, −7.05064254154, −6.59249513832, −5.07129023891, −3.93397206262, −2.78437107561, 2.78437107561, 3.93397206262, 5.07129023891, 6.59249513832, 7.05064254154, 8.06812083543, 8.46983732551, 9.22211166284, 9.60578416477, 10.9444831981, 11.3928378246, 11.6603604664, 12.5072882973, 13.1371488923, 14.1312186687, 14.6679752241, 15.1376917508, 15.5315873680, 16.1785116913, 16.8782224727, 17.3563779162, 17.9582356663, 18.5412106719, 19.1297294501, 19.5706470116

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