Properties

Label 4-5769-1.1-c1e2-0-2
Degree $4$
Conductor $5769$
Sign $1$
Analytic cond. $0.367836$
Root an. cond. $0.778778$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4·4-s − 4·5-s + 9·6-s − 3·7-s − 3·8-s + 6·9-s + 12·10-s − 6·11-s − 12·12-s − 10·13-s + 9·14-s + 12·15-s + 3·16-s + 3·17-s − 18·18-s + 2·19-s − 16·20-s + 9·21-s + 18·22-s − 5·23-s + 9·24-s + 5·25-s + 30·26-s − 9·27-s − 12·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 1.78·5-s + 3.67·6-s − 1.13·7-s − 1.06·8-s + 2·9-s + 3.79·10-s − 1.80·11-s − 3.46·12-s − 2.77·13-s + 2.40·14-s + 3.09·15-s + 3/4·16-s + 0.727·17-s − 4.24·18-s + 0.458·19-s − 3.57·20-s + 1.96·21-s + 3.83·22-s − 1.04·23-s + 1.83·24-s + 25-s + 5.88·26-s − 1.73·27-s − 2.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5769\)    =    \(3^{2} \cdot 641\)
Sign: $1$
Analytic conductor: \(0.367836\)
Root analytic conductor: \(0.778778\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5769,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_2$ \( 1 + p T + p T^{2} \)
641$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 35 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 13 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T - 23 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 7 T + 49 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$D_{4}$ \( 1 + 6 T + 21 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 63 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 103 T^{2} + p^{2} T^{4} \)
79$C_4$ \( 1 + 9 T + 19 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 9 T + 165 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 119 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−17.7887398412, −17.4073818074, −16.7698979160, −16.5992262701, −16.0339794570, −15.6861672048, −15.3209896869, −14.5970861973, −13.4892470270, −12.5592118667, −12.3444692787, −11.9700490053, −11.5167000126, −10.5414661090, −10.2355035931, −9.99286055792, −9.42410516766, −8.42168990912, −7.69273001323, −7.45836064364, −7.11204416980, −5.88561850899, −5.18481986248, −4.35807277813, −2.92987691163, 0, 0, 2.92987691163, 4.35807277813, 5.18481986248, 5.88561850899, 7.11204416980, 7.45836064364, 7.69273001323, 8.42168990912, 9.42410516766, 9.99286055792, 10.2355035931, 10.5414661090, 11.5167000126, 11.9700490053, 12.3444692787, 12.5592118667, 13.4892470270, 14.5970861973, 15.3209896869, 15.6861672048, 16.0339794570, 16.5992262701, 16.7698979160, 17.4073818074, 17.7887398412

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