Properties

Degree 4
Conductor 68209
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 − 3·3-s + 4-s − 5·5-s + 6·6-s − 5·7-s + 2·9-s + 10·10-s − 6·11-s − 3·12-s − 4·13-s + 10·14-s + 15·15-s + 16-s − 3·17-s − 4·18-s − 4·19-s − 5·20-s + 15·21-s + 12·22-s + 12·25-s + 8·26-s + 6·27-s − 5·28-s − 2·29-s − 30·30-s − 8·31-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.23·5-s + 2.44·6-s − 1.88·7-s + 2/3·9-s + 3.16·10-s − 1.80·11-s − 0.866·12-s − 1.10·13-s + 2.67·14-s + 3.87·15-s + 1/4·16-s − 0.727·17-s − 0.942·18-s − 0.917·19-s − 1.11·20-s + 3.27·21-s + 2.55·22-s + 12/5·25-s + 1.56·26-s + 1.15·27-s − 0.944·28-s − 0.371·29-s − 5.47·30-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 68209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 68209 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(68209\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{68209} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(4,\ 68209,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 68209$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 68209$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad68209$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 164 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$D_{4}$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 13 T + 131 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 68 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 9 T + 55 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 7 T + 49 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 9 T + 74 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 5 T + 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
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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

−15.3542799327, −14.9129297544, −14.5483359588, −13.3664072847, −13.0527359114, −12.6398187873, −12.2545035496, −11.946712512, −11.2763232003, −11.0289793245, −10.6705118755, −10.0695846662, −9.75194230475, −9.02567236211, −8.70142897324, −7.94845227303, −7.72905993567, −7.16530808388, −6.57034815906, −6.09903161561, −5.39873420863, −4.80461964205, −4.1781426897, −3.27159114304, −2.71913210686, 0, 0, 0, 2.71913210686, 3.27159114304, 4.1781426897, 4.80461964205, 5.39873420863, 6.09903161561, 6.57034815906, 7.16530808388, 7.72905993567, 7.94845227303, 8.70142897324, 9.02567236211, 9.75194230475, 10.0695846662, 10.6705118755, 11.0289793245, 11.2763232003, 11.946712512, 12.2545035496, 12.6398187873, 13.0527359114, 13.3664072847, 14.5483359588, 14.9129297544, 15.3542799327

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