Properties

Degree 4
Conductor 41411
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 − 6·5-s + 8·6-s − 4·7-s + 7·9-s + 12·10-s − 8·11-s − 4·12-s − 13-s + 8·14-s + 24·15-s + 16-s − 17-s − 14·18-s − 8·19-s − 6·20-s + 16·21-s + 16·22-s − 2·23-s + 17·25-s + 2·26-s − 4·27-s − 4·28-s − 2·29-s − 48·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 1/2·4-s − 2.68·5-s + 3.26·6-s − 1.51·7-s + 7/3·9-s + 3.79·10-s − 2.41·11-s − 1.15·12-s − 0.277·13-s + 2.13·14-s + 6.19·15-s + 1/4·16-s − 0.242·17-s − 3.29·18-s − 1.83·19-s − 1.34·20-s + 3.49·21-s + 3.41·22-s − 0.417·23-s + 17/5·25-s + 0.392·26-s − 0.769·27-s − 0.755·28-s − 0.371·29-s − 8.76·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 41411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 41411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(41411\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{41411} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 41411,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 41411$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 41411$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad41411$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 97 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 8 T + 3 p T^{2} + 8 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T - 11 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 13 T + 89 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 28 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 13 T + 145 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + T + 46 T^{2} + p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 10 T + 25 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T - 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−15.7803069871, −15.5908650281, −15.202908983, −14.620817694, −13.3935445545, −12.981665964, −12.538791339, −12.3491097666, −11.7440610738, −11.3765852779, −10.9234100313, −10.5836222993, −10.1738461086, −9.71373056287, −8.78822225726, −8.36870668818, −7.91007009165, −7.48838834203, −6.89672448604, −6.20207696938, −5.87016165543, −4.77266434696, −4.6717070966, −3.62943029225, −2.86913727265, 0, 0, 0, 2.86913727265, 3.62943029225, 4.6717070966, 4.77266434696, 5.87016165543, 6.20207696938, 6.89672448604, 7.48838834203, 7.91007009165, 8.36870668818, 8.78822225726, 9.71373056287, 10.1738461086, 10.5836222993, 10.9234100313, 11.3765852779, 11.7440610738, 12.3491097666, 12.538791339, 12.981665964, 13.3935445545, 14.620817694, 15.202908983, 15.5908650281, 15.7803069871

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