Properties

Degree 4
Conductor $ 2^{2} \cdot 127 \cdot 197 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 2·5-s + 2·6-s + 8-s + 2·10-s − 5·11-s − 2·13-s + 4·15-s − 16-s − 4·17-s − 5·19-s + 5·22-s − 23-s − 2·24-s − 2·25-s + 2·26-s + 2·27-s − 10·29-s − 4·30-s − 3·31-s + 10·33-s + 4·34-s + 2·37-s + 5·38-s + 4·39-s − 2·40-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 0.894·5-s + 0.816·6-s + 0.353·8-s + 0.632·10-s − 1.50·11-s − 0.554·13-s + 1.03·15-s − 1/4·16-s − 0.970·17-s − 1.14·19-s + 1.06·22-s − 0.208·23-s − 0.408·24-s − 2/5·25-s + 0.392·26-s + 0.384·27-s − 1.85·29-s − 0.730·30-s − 0.538·31-s + 1.74·33-s + 0.685·34-s + 0.328·37-s + 0.811·38-s + 0.640·39-s − 0.316·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100076 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100076 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100076\)    =    \(2^{2} \cdot 127 \cdot 197\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100076} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 100076,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;127,\;197\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;127,\;197\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T + T^{2} \)
127$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 16 T + p T^{2} ) \)
197$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 24 T + p T^{2} ) \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 5 T + 23 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 53 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 2 T - 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 57 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 3 T + 21 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 77 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 108 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 7 T + p T^{2} + 7 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 7 T - 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−14.8643051132, −13.9459416523, −13.4258968078, −13.1522588322, −12.6172129676, −12.2716824811, −11.6531611049, −11.3301282100, −10.8984073682, −10.7371571465, −10.0661540924, −9.69371361769, −8.99204439595, −8.57812574813, −8.05115856226, −7.65911692338, −7.15880121588, −6.67556657027, −5.73212223416, −5.65952058234, −4.91590786631, −4.31414036577, −3.76757622262, −2.68842566549, −1.95069692694, 0, 0, 1.95069692694, 2.68842566549, 3.76757622262, 4.31414036577, 4.91590786631, 5.65952058234, 5.73212223416, 6.67556657027, 7.15880121588, 7.65911692338, 8.05115856226, 8.57812574813, 8.99204439595, 9.69371361769, 10.0661540924, 10.7371571465, 10.8984073682, 11.3301282100, 11.6531611049, 12.2716824811, 12.6172129676, 13.1522588322, 13.4258968078, 13.9459416523, 14.8643051132

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