Properties

Degree 4
Conductor $ 2 \cdot 33503 $
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 − 4·3-s + 4-s − 4·5-s + 8·6-s − 6·7-s + 2·8-s + 7·9-s + 8·10-s − 5·11-s − 4·12-s − 5·13-s + 12·14-s + 16·15-s − 3·16-s − 3·17-s − 14·18-s − 4·19-s − 4·20-s + 24·21-s + 10·22-s − 4·23-s − 8·24-s + 5·25-s + 10·26-s − 4·27-s − 6·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 3.26·6-s − 2.26·7-s + 0.707·8-s + 7/3·9-s + 2.52·10-s − 1.50·11-s − 1.15·12-s − 1.38·13-s + 3.20·14-s + 4.13·15-s − 3/4·16-s − 0.727·17-s − 3.29·18-s − 0.917·19-s − 0.894·20-s + 5.23·21-s + 2.13·22-s − 0.834·23-s − 1.63·24-s + 25-s + 1.96·26-s − 0.769·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 67006 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 67006 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(67006\)    =    \(2 \cdot 33503\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{67006} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 67006,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;33503\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;33503\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
33503$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 220 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$D_{4}$ \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 67 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T + 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T + 3 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - T + 109 T^{2} - p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 5 T + 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 7 T + 41 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 8 T + 29 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 13 T + 162 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−15.5539757525, −14.9247945101, −14.4802307604, −13.2963572858, −13.0223104262, −12.7189856225, −12.3110974627, −11.9192117907, −11.3388604242, −10.9029307898, −10.6770696698, −10.2126545831, −9.67329394846, −9.28582691549, −8.75799951087, −7.79401349774, −7.63946823261, −7.19024418931, −6.61023923575, −5.90067693882, −5.61109753087, −4.82623293883, −4.16864403375, −3.55241077581, −2.44340163261, 0, 0, 0, 2.44340163261, 3.55241077581, 4.16864403375, 4.82623293883, 5.61109753087, 5.90067693882, 6.61023923575, 7.19024418931, 7.63946823261, 7.79401349774, 8.75799951087, 9.28582691549, 9.67329394846, 10.2126545831, 10.6770696698, 10.9029307898, 11.3388604242, 11.9192117907, 12.3110974627, 12.7189856225, 13.0223104262, 13.2963572858, 14.4802307604, 14.9247945101, 15.5539757525

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