Properties

Degree 4
Conductor $ 3 \cdot 5 \cdot 23 \cdot 29 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 4·4-s − 3·5-s + 6·6-s − 5·7-s − 3·8-s + 4·9-s + 9·10-s − 7·11-s − 8·12-s − 4·13-s + 15·14-s + 6·15-s + 3·16-s − 17-s − 12·18-s − 2·19-s − 12·20-s + 10·21-s + 21·22-s − 7·23-s + 6·24-s + 2·25-s + 12·26-s − 5·27-s − 20·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s − 1.34·5-s + 2.44·6-s − 1.88·7-s − 1.06·8-s + 4/3·9-s + 2.84·10-s − 2.11·11-s − 2.30·12-s − 1.10·13-s + 4.00·14-s + 1.54·15-s + 3/4·16-s − 0.242·17-s − 2.82·18-s − 0.458·19-s − 2.68·20-s + 2.18·21-s + 4.47·22-s − 1.45·23-s + 1.22·24-s + 2/5·25-s + 2.35·26-s − 0.962·27-s − 3.77·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(10005\)    =    \(3 \cdot 5 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{10005} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 10005,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;23,\;29\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 6 T + p T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T - 8 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T - 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$D_{4}$ \( 1 + 5 T + 112 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \)
73$D_{4}$ \( 1 - 5 T + 56 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 18 T + 166 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 17 T + 188 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T - 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−17.0305853597, −16.6220522871, −16.3134047927, −15.8323722713, −15.4888692302, −15.3008590699, −14.1896249638, −13.1750842647, −12.8383387071, −12.4791925880, −11.9438562267, −11.2814501675, −10.6948802146, −10.1685117182, −9.85229374616, −9.62788712882, −8.63308358868, −8.04745725326, −7.59062894660, −7.18547537930, −6.39263081412, −5.70151241334, −4.75083207653, −3.77938542657, −2.61433593162, 0, 0, 2.61433593162, 3.77938542657, 4.75083207653, 5.70151241334, 6.39263081412, 7.18547537930, 7.59062894660, 8.04745725326, 8.63308358868, 9.62788712882, 9.85229374616, 10.1685117182, 10.6948802146, 11.2814501675, 11.9438562267, 12.4791925880, 12.8383387071, 13.1750842647, 14.1896249638, 15.3008590699, 15.4888692302, 15.8323722713, 16.3134047927, 16.6220522871, 17.0305853597

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