Properties

Label 4-1549-1.1-c1e2-0-0
Degree $4$
Conductor $1549$
Sign $-1$
Analytic cond. $0.0987655$
Root an. cond. $0.560597$
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 − 3·3-s − 2·4-s − 2·5-s + 3·6-s + 3·8-s + 2·9-s + 2·10-s + 6·12-s − 3·13-s + 6·15-s + 16-s + 2·17-s − 2·18-s + 4·20-s + 23-s − 9·24-s − 4·25-s + 3·26-s + 6·27-s − 6·30-s − 6·31-s − 2·32-s − 2·34-s − 4·36-s − 4·37-s + 9·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 4-s − 0.894·5-s + 1.22·6-s + 1.06·8-s + 2/3·9-s + 0.632·10-s + 1.73·12-s − 0.832·13-s + 1.54·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s + 0.894·20-s + 0.208·23-s − 1.83·24-s − 4/5·25-s + 0.588·26-s + 1.15·27-s − 1.09·30-s − 1.07·31-s − 0.353·32-s − 0.342·34-s − 2/3·36-s − 0.657·37-s + 1.44·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1549 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1549 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1549\)
Sign: $-1$
Analytic conductor: \(0.0987655\)
Root analytic conductor: \(0.560597\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1549,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad1549$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 21 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 9 T + 49 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 64 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 90 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T - 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 52 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 11 T + 157 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
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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

−19.3931496237, −18.7163139490, −18.4376751214, −17.7310764208, −17.3838370289, −17.0561631337, −16.4955242265, −16.0262884127, −15.2206098305, −14.5831700849, −13.9300366923, −13.2018471855, −12.3739347129, −12.0648653317, −11.3422266511, −10.9915420748, −10.0549571708, −9.63822321365, −8.70090650648, −8.09089382289, −7.31902782095, −6.35562373870, −5.31766035872, −4.92416847963, −3.74319017368, 0, 3.74319017368, 4.92416847963, 5.31766035872, 6.35562373870, 7.31902782095, 8.09089382289, 8.70090650648, 9.63822321365, 10.0549571708, 10.9915420748, 11.3422266511, 12.0648653317, 12.3739347129, 13.2018471855, 13.9300366923, 14.5831700849, 15.2206098305, 16.0262884127, 16.4955242265, 17.0561631337, 17.3838370289, 17.7310764208, 18.4376751214, 18.7163139490, 19.3931496237

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