Properties

Degree 4
Conductor $ 67^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4·4-s − 6·5-s + 9·6-s − 7-s − 3·8-s + 2·9-s + 18·10-s − 12·12-s − 7·13-s + 3·14-s + 18·15-s + 3·16-s − 6·17-s − 6·18-s + 19-s − 24·20-s + 3·21-s + 6·23-s + 9·24-s + 17·25-s + 21·26-s + 6·27-s − 4·28-s − 6·29-s − 54·30-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 2.68·5-s + 3.67·6-s − 0.377·7-s − 1.06·8-s + 2/3·9-s + 5.69·10-s − 3.46·12-s − 1.94·13-s + 0.801·14-s + 4.64·15-s + 3/4·16-s − 1.45·17-s − 1.41·18-s + 0.229·19-s − 5.36·20-s + 0.654·21-s + 1.25·23-s + 1.83·24-s + 17/5·25-s + 4.11·26-s + 1.15·27-s − 0.755·28-s − 1.11·29-s − 9.85·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4489\)    =    \(67^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4489} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 4489,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 67$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 67$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad67$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + T + 63 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 83 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 15 T + 149 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 173 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 15 T + 161 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 173 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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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

−18.0130037204, −17.6190626226, −17.2015751856, −16.6641079682, −16.6161348426, −15.7989628933, −15.5362758792, −14.9152377661, −14.4246188389, −12.8162016815, −12.6716190363, −11.8357455267, −11.5600106704, −11.1478021758, −10.8565129088, −9.92743865939, −9.41920432229, −8.64088142348, −8.20345258968, −7.35273556159, −7.26795772043, −6.3647069766, −5.11384122877, −4.54764937214, −3.25253030363, 0, 0, 3.25253030363, 4.54764937214, 5.11384122877, 6.3647069766, 7.26795772043, 7.35273556159, 8.20345258968, 8.64088142348, 9.41920432229, 9.92743865939, 10.8565129088, 11.1478021758, 11.5600106704, 11.8357455267, 12.6716190363, 12.8162016815, 14.4246188389, 14.9152377661, 15.5362758792, 15.7989628933, 16.6161348426, 16.6641079682, 17.2015751856, 17.6190626226, 18.0130037204

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