Properties

Degree 4
Conductor $ 3^{2} \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 2·4-s − 6-s + 2·7-s − 5·8-s − 4·11-s + 2·12-s − 5·13-s − 2·14-s + 5·16-s + 4·17-s − 8·19-s + 2·21-s + 4·22-s + 4·23-s − 5·24-s + 5·25-s + 5·26-s − 27-s + 4·28-s + 8·29-s − 6·31-s − 10·32-s − 4·33-s − 4·34-s + 6·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 4-s − 0.408·6-s + 0.755·7-s − 1.76·8-s − 1.20·11-s + 0.577·12-s − 1.38·13-s − 0.534·14-s + 5/4·16-s + 0.970·17-s − 1.83·19-s + 0.436·21-s + 0.852·22-s + 0.834·23-s − 1.02·24-s + 25-s + 0.980·26-s − 0.192·27-s + 0.755·28-s + 1.48·29-s − 1.07·31-s − 1.76·32-s − 0.696·33-s − 0.685·34-s + 0.986·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3249\)    =    \(3^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{57} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3249,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.663889$
$L(\frac12)$  $\approx$  $0.663889$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_2$ \( 1 - T + T^{2} \)
19$C_2$ \( 1 + 8 T + p T^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.35860856735610309898704050614, −14.95904509442818236902213106077, −14.50373110497224869643878491657, −14.34373906974706481615562195229, −12.98621670853570510070654685970, −12.67371856647351296888438461030, −12.20507992973222323767465292571, −11.42171316795461322732629094461, −10.75341689053599515996358739339, −10.48221575496679757289342798633, −9.513488209222794924894177986716, −9.097133540483271614089070009842, −8.262668132440760944071462396690, −7.894860116879307797435495500975, −7.18240199484817905418660626402, −6.38696664656561744722230314604, −5.48594606438498548166885130784, −4.60961511505194332926976804502, −2.94946291741863806250225006895, −2.40818147527031805418881116977, 2.40818147527031805418881116977, 2.94946291741863806250225006895, 4.60961511505194332926976804502, 5.48594606438498548166885130784, 6.38696664656561744722230314604, 7.18240199484817905418660626402, 7.894860116879307797435495500975, 8.262668132440760944071462396690, 9.097133540483271614089070009842, 9.513488209222794924894177986716, 10.48221575496679757289342798633, 10.75341689053599515996358739339, 11.42171316795461322732629094461, 12.20507992973222323767465292571, 12.67371856647351296888438461030, 12.98621670853570510070654685970, 14.34373906974706481615562195229, 14.50373110497224869643878491657, 14.95904509442818236902213106077, 15.35860856735610309898704050614

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