Properties

Label 4-1239-1.1-c1e2-0-0
Degree $4$
Conductor $1239$
Sign $1$
Analytic cond. $0.0789997$
Root an. cond. $0.530159$
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  − 4-s − 5-s − 2·7-s − 2·9-s + 4·11-s − 2·13-s − 3·16-s + 2·17-s − 3·19-s + 20-s + 6·23-s + 3·25-s − 3·27-s + 2·28-s + 3·29-s + 4·31-s + 2·35-s + 2·36-s − 2·37-s + 3·41-s − 4·44-s + 2·45-s − 4·47-s − 4·49-s + 2·52-s − 3·53-s − 4·55-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 1.20·11-s − 0.554·13-s − 3/4·16-s + 0.485·17-s − 0.688·19-s + 0.223·20-s + 1.25·23-s + 3/5·25-s − 0.577·27-s + 0.377·28-s + 0.557·29-s + 0.718·31-s + 0.338·35-s + 1/3·36-s − 0.328·37-s + 0.468·41-s − 0.603·44-s + 0.298·45-s − 0.583·47-s − 4/7·49-s + 0.277·52-s − 0.412·53-s − 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1239 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1239 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1239\)    =    \(3 \cdot 7 \cdot 59\)
Sign: $1$
Analytic conductor: \(0.0789997\)
Root analytic conductor: \(0.530159\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1239,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4698218148\)
\(L(\frac12)\) \(\approx\) \(0.4698218148\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
59$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$D_{4}$ \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$D_{4}$ \( 1 - 17 T + 194 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T - 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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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

−19.4454481159, −19.2622361255, −18.7570685606, −17.8442438628, −17.4316697374, −16.8356170527, −16.4586392021, −15.7396638868, −14.9443883737, −14.6667898472, −13.9023464480, −13.3904914266, −12.6062840522, −12.1380141538, −11.4236905126, −10.8481273875, −9.86871522130, −9.25080350873, −8.77527783532, −7.92545753803, −6.85133126119, −6.37047224055, −5.11557499710, −4.18132838118, −3.03733488669, 3.03733488669, 4.18132838118, 5.11557499710, 6.37047224055, 6.85133126119, 7.92545753803, 8.77527783532, 9.25080350873, 9.86871522130, 10.8481273875, 11.4236905126, 12.1380141538, 12.6062840522, 13.3904914266, 13.9023464480, 14.6667898472, 14.9443883737, 15.7396638868, 16.4586392021, 16.8356170527, 17.4316697374, 17.8442438628, 18.7570685606, 19.2622361255, 19.4454481159

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