Properties

Degree 4
Conductor $ 7 \cdot 14293 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s − 2·5-s − 2·6-s − 3·7-s + 4·8-s − 4·9-s + 4·10-s − 4·11-s + 6·13-s + 6·14-s − 2·15-s − 4·16-s + 17-s + 8·18-s + 5·19-s − 3·21-s + 8·22-s + 6·23-s + 4·24-s − 4·25-s − 12·26-s − 6·27-s + 7·29-s + 4·30-s − 6·31-s − 4·33-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s − 0.894·5-s − 0.816·6-s − 1.13·7-s + 1.41·8-s − 4/3·9-s + 1.26·10-s − 1.20·11-s + 1.66·13-s + 1.60·14-s − 0.516·15-s − 16-s + 0.242·17-s + 1.88·18-s + 1.14·19-s − 0.654·21-s + 1.70·22-s + 1.25·23-s + 0.816·24-s − 4/5·25-s − 2.35·26-s − 1.15·27-s + 1.29·29-s + 0.730·30-s − 1.07·31-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100051 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100051 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100051\)    =    \(7 \cdot 14293\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100051} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 100051,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;14293\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;14293\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
14293$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 106 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$D_{4}$ \( 1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 5 T + 10 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 9 T + 77 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T + 41 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 11 T + 67 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 11 T + 87 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 108 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - T + 83 T^{2} - p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - T + 29 T^{2} - p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$D_{4}$ \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.2363773761, −13.8283084955, −13.3648427865, −13.0296013319, −12.73370512, −11.8559732863, −11.5066112169, −11.0800640796, −10.5861678512, −10.1110360819, −9.66956971282, −9.10593387724, −8.8139394137, −8.5382728788, −8.00886720957, −7.59598497919, −7.27231276767, −6.25083425741, −5.84857455155, −5.22916051104, −4.4465869737, −3.54145331701, −3.29217649155, −2.59013213873, −1.01015056354, 0, 1.01015056354, 2.59013213873, 3.29217649155, 3.54145331701, 4.4465869737, 5.22916051104, 5.84857455155, 6.25083425741, 7.27231276767, 7.59598497919, 8.00886720957, 8.5382728788, 8.8139394137, 9.10593387724, 9.66956971282, 10.1110360819, 10.5861678512, 11.0800640796, 11.5066112169, 11.8559732863, 12.73370512, 13.0296013319, 13.3648427865, 13.8283084955, 14.2363773761

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