Properties

Label 4-5317-1.1-c1e2-0-0
Degree $4$
Conductor $5317$
Sign $-1$
Analytic cond. $0.339016$
Root an. cond. $0.763053$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 2·5-s + 6-s − 3·7-s − 3·8-s − 4·9-s + 2·10-s + 11-s − 12-s − 13-s + 3·14-s + 2·15-s + 16-s + 5·17-s + 4·18-s − 3·19-s − 2·20-s + 3·21-s − 22-s − 3·23-s + 3·24-s + 26-s + 6·27-s − 3·28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.13·7-s − 1.06·8-s − 4/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.801·14-s + 0.516·15-s + 1/4·16-s + 1.21·17-s + 0.942·18-s − 0.688·19-s − 0.447·20-s + 0.654·21-s − 0.213·22-s − 0.625·23-s + 0.612·24-s + 0.196·26-s + 1.15·27-s − 0.566·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5317 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5317 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5317\)    =    \(13 \cdot 409\)
Sign: $-1$
Analytic conductor: \(0.339016\)
Root analytic conductor: \(0.763053\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5317,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
409$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 22 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T - 14 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 5 T + p T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T - 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 35 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T - 83 T^{2} + p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$D_{4}$ \( 1 + 7 T + 91 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
73$D_{4}$ \( 1 + 7 T + 52 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
89$D_{4}$ \( 1 + 9 T + 138 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 159 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−17.6033531041, −17.0618994787, −16.8345694541, −16.1957281256, −15.7141479146, −15.4517109620, −14.6188487824, −14.2259796645, −13.6324371359, −12.5017538057, −12.3524504777, −11.8848819727, −11.3265510800, −10.8670173111, −9.96924656010, −9.68160347429, −8.79241923217, −8.40578918090, −7.75330917506, −6.84382316573, −6.20517744958, −5.86154945091, −4.70265601650, −3.44595532193, −2.83428800912, 0, 2.83428800912, 3.44595532193, 4.70265601650, 5.86154945091, 6.20517744958, 6.84382316573, 7.75330917506, 8.40578918090, 8.79241923217, 9.68160347429, 9.96924656010, 10.8670173111, 11.3265510800, 11.8848819727, 12.3524504777, 12.5017538057, 13.6324371359, 14.2259796645, 14.6188487824, 15.4517109620, 15.7141479146, 16.1957281256, 16.8345694541, 17.0618994787, 17.6033531041

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