Properties

Degree 4
Conductor 61553
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 − 3·3-s + 4·4-s − 5·5-s + 9·6-s − 6·7-s − 3·8-s + 4·9-s + 15·10-s − 11-s − 12·12-s − 4·13-s + 18·14-s + 15·15-s + 3·16-s − 6·17-s − 12·18-s − 8·19-s − 20·20-s + 18·21-s + 3·22-s − 6·23-s + 9·24-s + 11·25-s + 12·26-s − 6·27-s − 24·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 2.23·5-s + 3.67·6-s − 2.26·7-s − 1.06·8-s + 4/3·9-s + 4.74·10-s − 0.301·11-s − 3.46·12-s − 1.10·13-s + 4.81·14-s + 3.87·15-s + 3/4·16-s − 1.45·17-s − 2.82·18-s − 1.83·19-s − 4.47·20-s + 3.92·21-s + 0.639·22-s − 1.25·23-s + 1.83·24-s + 11/5·25-s + 2.35·26-s − 1.15·27-s − 4.53·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 61553 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 61553 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(61553\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{61553} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 61553,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 61553$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 61553$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad61553$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 246 T + p T^{2} ) \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 + 6 T + 3 p T^{2} + 6 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T - 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$D_{4}$ \( 1 + T + 81 T^{2} + p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T + 81 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + T + 57 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 7 T + 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 10 T + 185 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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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

−15.4433512707, −15.1741237701, −14.8203394574, −13.6318472059, −12.9807482343, −12.7300538848, −12.2465802196, −11.9345413319, −11.4306443861, −10.9941975493, −10.611334399, −10.1568094247, −9.71550375462, −9.22282764431, −8.75017973488, −8.21320460921, −7.61858596394, −7.26600228894, −6.74168778016, −6.22125539553, −5.76854289515, −4.70883671738, −4.02331398735, −3.57414344466, −2.33412559016, 0, 0, 0, 2.33412559016, 3.57414344466, 4.02331398735, 4.70883671738, 5.76854289515, 6.22125539553, 6.74168778016, 7.26600228894, 7.61858596394, 8.21320460921, 8.75017973488, 9.22282764431, 9.71550375462, 10.1568094247, 10.611334399, 10.9941975493, 11.4306443861, 11.9345413319, 12.2465802196, 12.7300538848, 12.9807482343, 13.6318472059, 14.8203394574, 15.1741237701, 15.4433512707

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