Properties

Label 4-249-1.1-c1e2-0-0
Degree $4$
Conductor $249$
Sign $1$
Analytic cond. $0.0158764$
Root an. cond. $0.354967$
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  − 2·2-s − 2·3-s + 4-s + 4·6-s − 7-s + 4·9-s + 11-s − 2·12-s + 2·14-s + 16-s − 17-s − 8·18-s − 6·19-s + 2·21-s − 2·22-s − 2·25-s − 5·27-s − 28-s + 3·29-s + 3·31-s + 2·32-s − 2·33-s + 2·34-s + 4·36-s − 5·37-s + 12·38-s + 4·41-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s − 0.377·7-s + 4/3·9-s + 0.301·11-s − 0.577·12-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 1.88·18-s − 1.37·19-s + 0.436·21-s − 0.426·22-s − 2/5·25-s − 0.962·27-s − 0.188·28-s + 0.557·29-s + 0.538·31-s + 0.353·32-s − 0.348·33-s + 0.342·34-s + 2/3·36-s − 0.821·37-s + 1.94·38-s + 0.624·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(249\)    =    \(3 \cdot 83\)
Sign: $1$
Analytic conductor: \(0.0158764\)
Root analytic conductor: \(0.354967\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 249,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1315495070\)
\(L(\frac12)\) \(\approx\) \(0.1315495070\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T - 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 13 T + 106 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T + 48 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$D_{4}$ \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
97$D_{4}$ \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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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

−19.9583362020, −19.2532545300, −18.8911406702, −18.3038693797, −17.5495656198, −17.3478671179, −16.7780994692, −15.9022638292, −15.5120887195, −14.4574259478, −13.4301254485, −12.5870403887, −12.0374010604, −11.0180573664, −10.4121692020, −9.68468363219, −8.95320072320, −8.02996189451, −6.82879614383, −6.02680203091, −4.45419105482, 4.45419105482, 6.02680203091, 6.82879614383, 8.02996189451, 8.95320072320, 9.68468363219, 10.4121692020, 11.0180573664, 12.0374010604, 12.5870403887, 13.4301254485, 14.4574259478, 15.5120887195, 15.9022638292, 16.7780994692, 17.3478671179, 17.5495656198, 18.3038693797, 18.8911406702, 19.2532545300, 19.9583362020

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