Properties

Degree 4
Conductor 62233
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 4·3-s + 4·4-s − 3·5-s + 12·6-s − 5·7-s − 3·8-s + 7·9-s + 9·10-s − 4·11-s − 16·12-s − 6·13-s + 15·14-s + 12·15-s + 3·16-s − 6·17-s − 21·18-s − 5·19-s − 12·20-s + 20·21-s + 12·22-s + 12·24-s + 25-s + 18·26-s − 4·27-s − 20·28-s − 5·29-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 2·4-s − 1.34·5-s + 4.89·6-s − 1.88·7-s − 1.06·8-s + 7/3·9-s + 2.84·10-s − 1.20·11-s − 4.61·12-s − 1.66·13-s + 4.00·14-s + 3.09·15-s + 3/4·16-s − 1.45·17-s − 4.94·18-s − 1.14·19-s − 2.68·20-s + 4.36·21-s + 2.55·22-s + 2.44·24-s + 1/5·25-s + 3.53·26-s − 0.769·27-s − 3.77·28-s − 0.928·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 62233 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 62233 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(62233\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{62233} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 62233,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 62233$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 62233$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad62233$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 146 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T + 41 T^{2} + p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 15 T + 131 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 10 T + 103 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 5 T + 114 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
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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

−15.603260606, −15.0401891794, −14.7691158996, −13.5004191479, −13.0994143524, −12.6280069341, −12.2112481148, −12.0124360748, −11.3229292907, −10.9651598234, −10.5789156256, −10.2073323014, −9.77767327962, −9.31983445962, −8.69767416992, −8.27309005102, −7.54982726691, −7.12745445612, −6.73390487304, −6.25359064986, −5.4404040264, −5.10272616484, −4.20719486567, −3.36502681375, −2.25072988347, 0, 0, 0, 2.25072988347, 3.36502681375, 4.20719486567, 5.10272616484, 5.4404040264, 6.25359064986, 6.73390487304, 7.12745445612, 7.54982726691, 8.27309005102, 8.69767416992, 9.31983445962, 9.77767327962, 10.2073323014, 10.5789156256, 10.9651598234, 11.3229292907, 12.0124360748, 12.2112481148, 12.6280069341, 13.0994143524, 13.5004191479, 14.7691158996, 15.0401891794, 15.603260606

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