Properties

Degree 4
Conductor $ 2^{2} \cdot 2297 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·3-s − 5·5-s + 4·6-s − 4·7-s + 8-s + 7·9-s + 5·10-s − 4·11-s − 13-s + 4·14-s + 20·15-s − 16-s + 3·17-s − 7·18-s − 6·19-s + 16·21-s + 4·22-s + 2·23-s − 4·24-s + 12·25-s + 26-s − 4·27-s − 2·29-s − 20·30-s − 8·31-s + 16·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s − 2.23·5-s + 1.63·6-s − 1.51·7-s + 0.353·8-s + 7/3·9-s + 1.58·10-s − 1.20·11-s − 0.277·13-s + 1.06·14-s + 5.16·15-s − 1/4·16-s + 0.727·17-s − 1.64·18-s − 1.37·19-s + 3.49·21-s + 0.852·22-s + 0.417·23-s − 0.816·24-s + 12/5·25-s + 0.196·26-s − 0.769·27-s − 0.371·29-s − 3.65·30-s − 1.43·31-s + 2.78·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9188 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9188 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9188\)    =    \(2^{2} \cdot 2297\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{9188} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 9188,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;2297\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;2297\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + T^{2} \)
2297$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 54 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 6 T + 15 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 112 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
67$D_{4}$ \( 1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$D_{4}$ \( 1 + 7 T + 38 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
83$D_{4}$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 9 T + 29 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 24 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−17.0809907408, −16.6383040516, −16.3409815327, −15.883043108, −15.7035083168, −15.0466474344, −14.4624516952, −13.2744035267, −12.8292126508, −12.3832383276, −12.1570320992, −11.436708553, −11.1113091071, −10.5766469092, −10.3046246222, −9.4854849051, −8.64652314095, −8.0763767815, −7.32193870605, −7.00138962018, −6.2293863696, −5.50905912284, −4.907456964, −4.02375394746, −3.22399558858, 0, 0, 3.22399558858, 4.02375394746, 4.907456964, 5.50905912284, 6.2293863696, 7.00138962018, 7.32193870605, 8.0763767815, 8.64652314095, 9.4854849051, 10.3046246222, 10.5766469092, 11.1113091071, 11.436708553, 12.1570320992, 12.3832383276, 12.8292126508, 13.2744035267, 14.4624516952, 15.0466474344, 15.7035083168, 15.883043108, 16.3409815327, 16.6383040516, 17.0809907408

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