Properties

Degree 4
Conductor $ 149 \cdot 673 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 11-s − 3·13-s − 4·16-s + 2·17-s − 6·19-s − 5·23-s − 6·25-s + 9·29-s − 14·31-s + 3·37-s − 5·41-s + 3·43-s − 3·47-s − 10·49-s − 5·53-s − 59-s − 9·61-s + 5·67-s − 71-s + 73-s + 4·79-s − 5·81-s + 25·83-s + 15·89-s + 8·97-s − 2·99-s + ⋯
L(s)  = 1  + 2/3·9-s − 0.301·11-s − 0.832·13-s − 16-s + 0.485·17-s − 1.37·19-s − 1.04·23-s − 6/5·25-s + 1.67·29-s − 2.51·31-s + 0.493·37-s − 0.780·41-s + 0.457·43-s − 0.437·47-s − 1.42·49-s − 0.686·53-s − 0.130·59-s − 1.15·61-s + 0.610·67-s − 0.118·71-s + 0.117·73-s + 0.450·79-s − 5/9·81-s + 2.74·83-s + 1.58·89-s + 0.812·97-s − 0.201·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100277 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100277 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100277\)    =    \(149 \cdot 673\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100277} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 100277,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{149,\;673\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{149,\;673\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad149$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 15 T + p T^{2} ) \)
673$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
3$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$D_{4}$ \( 1 - 3 T - 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \)
47$D_{4}$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + T + 115 T^{2} + p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 9 T + 48 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 5 T + 42 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
89$D_{4}$ \( 1 - 15 T + 188 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 6 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−14.3030475325, −13.7986985974, −13.4064840503, −12.9060928955, −12.5570236502, −12.1169932511, −11.7495978299, −11.1044707226, −10.6838747314, −10.3226983121, −9.70660404462, −9.45143118936, −8.92130387039, −8.18063269817, −7.88718255024, −7.39440000557, −6.75007466811, −6.34910746544, −5.79044547331, −4.95464493499, −4.63452670837, −3.96231625325, −3.31474041167, −2.25445272523, −1.83725972767, 0, 1.83725972767, 2.25445272523, 3.31474041167, 3.96231625325, 4.63452670837, 4.95464493499, 5.79044547331, 6.34910746544, 6.75007466811, 7.39440000557, 7.88718255024, 8.18063269817, 8.92130387039, 9.45143118936, 9.70660404462, 10.3226983121, 10.6838747314, 11.1044707226, 11.7495978299, 12.1169932511, 12.5570236502, 12.9060928955, 13.4064840503, 13.7986985974, 14.3030475325

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