Properties

Degree 4
Conductor $ 241 \cdot 269 $
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 − 5·5-s + 12·6-s − 2·7-s − 3·8-s + 7·9-s + 15·10-s − 4·11-s − 16·12-s − 2·13-s + 6·14-s + 20·15-s + 3·16-s − 9·17-s − 21·18-s − 6·19-s − 20·20-s + 8·21-s + 12·22-s − 5·23-s + 12·24-s + 10·25-s + 6·26-s − 4·27-s − 8·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.30·3-s + 2·4-s − 2.23·5-s + 4.89·6-s − 0.755·7-s − 1.06·8-s + 7/3·9-s + 4.74·10-s − 1.20·11-s − 4.61·12-s − 0.554·13-s + 1.60·14-s + 5.16·15-s + 3/4·16-s − 2.18·17-s − 4.94·18-s − 1.37·19-s − 4.47·20-s + 1.74·21-s + 2.55·22-s − 1.04·23-s + 2.44·24-s + 2·25-s + 1.17·26-s − 0.769·27-s − 1.51·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 64829 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 64829 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(64829\)    =    \(241 \cdot 269\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{64829} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 64829,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{241,\;269\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{241,\;269\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad241$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 13 T + p T^{2} ) \)
269$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 16 T + p T^{2} ) \)
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$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 9 T + 49 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 5 T + 45 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 10 T + 86 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 3 T - 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T - 11 T^{2} + p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
59$D_{4}$ \( 1 + 12 T + 119 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - T + 19 T^{2} - p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 133 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 3 T + 154 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + T + 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−15.4588659626, −15.2055477727, −14.6940690131, −13.4755808766, −13.0654587079, −12.5605510994, −12.2005796808, −11.7315724167, −11.3202761751, −10.9968384216, −10.6955866531, −10.3389600884, −9.68128752088, −9.15152591072, −8.5648557915, −8.16611523791, −7.6643552486, −7.26639806897, −6.63831763944, −6.2252734252, −5.51624597161, −4.80392588876, −4.24091311912, −3.51463347621, −2.12870515047, 0, 0, 0, 2.12870515047, 3.51463347621, 4.24091311912, 4.80392588876, 5.51624597161, 6.2252734252, 6.63831763944, 7.26639806897, 7.6643552486, 8.16611523791, 8.5648557915, 9.15152591072, 9.68128752088, 10.3389600884, 10.6955866531, 10.9968384216, 11.3202761751, 11.7315724167, 12.2005796808, 12.5605510994, 13.0654587079, 13.4755808766, 14.6940690131, 15.2055477727, 15.4588659626

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