Properties

Degree 4
Conductor 66161
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s − 4·5-s + 6·6-s − 7·7-s + 4·8-s + 3·9-s + 8·10-s − 4·11-s − 5·13-s + 14·14-s + 12·15-s − 4·16-s − 17-s − 6·18-s − 8·19-s + 21·21-s + 8·22-s + 3·23-s − 12·24-s + 6·25-s + 10·26-s − 4·29-s − 24·30-s − 12·31-s + 12·33-s + 2·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s − 1.78·5-s + 2.44·6-s − 2.64·7-s + 1.41·8-s + 9-s + 2.52·10-s − 1.20·11-s − 1.38·13-s + 3.74·14-s + 3.09·15-s − 16-s − 0.242·17-s − 1.41·18-s − 1.83·19-s + 4.58·21-s + 1.70·22-s + 0.625·23-s − 2.44·24-s + 6/5·25-s + 1.96·26-s − 0.742·29-s − 4.38·30-s − 2.15·31-s + 2.08·33-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 66161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 66161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(66161\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{66161} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 66161,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 66161$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 66161$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad66161$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 478 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$D_{4}$ \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T - 20 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 88 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 5 T + 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 49 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 46 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 35 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 15 T + 127 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 44 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$D_{4}$ \( 1 + 10 T + 52 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 11 T + 50 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
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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

−15.3135897601, −15.0082877926, −14.4461302909, −13.3040142846, −13.0552256222, −12.8982172409, −12.2836741392, −12.0493264825, −11.3825630596, −10.9043412183, −10.5013558824, −10.1999188836, −9.61748802, −9.26402369338, −8.66176369033, −8.22505394688, −7.51041080144, −7.06876650234, −6.80557895821, −5.9862874013, −5.42793031195, −4.85321776881, −4.04062925171, −3.57160188752, −2.58044205646, 0, 0, 0, 2.58044205646, 3.57160188752, 4.04062925171, 4.85321776881, 5.42793031195, 5.9862874013, 6.80557895821, 7.06876650234, 7.51041080144, 8.22505394688, 8.66176369033, 9.26402369338, 9.61748802, 10.1999188836, 10.5013558824, 10.9043412183, 11.3825630596, 12.0493264825, 12.2836741392, 12.8982172409, 13.0552256222, 13.3040142846, 14.4461302909, 15.0082877926, 15.3135897601

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