Properties

Degree 4
Conductor $ 3^{4} \cdot 5 \cdot 13 \cdot 19 $
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-s + 3-s − 2·4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s − 6·11-s − 2·12-s + 13-s − 15-s + 16-s + 4·17-s − 18-s + 2·20-s + 6·22-s + 6·23-s + 3·24-s − 4·25-s − 26-s + 27-s + 2·29-s + 30-s − 4·31-s − 2·32-s − 6·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s − 0.577·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.447·20-s + 1.27·22-s + 1.25·23-s + 0.612·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s − 0.718·31-s − 0.353·32-s − 1.04·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100035 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100035 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100035\)    =    \(3^{4} \cdot 5 \cdot 13 \cdot 19\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100035} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 100035,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;13,\;19\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;5,\;13,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_4$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 7 T + 24 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$D_{4}$ \( 1 - T - 74 T^{2} - p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T - 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 105 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 17 T + 196 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
89$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 5 T + 178 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
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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

−14.3182182802, −13.6791203364, −13.3562933455, −13.1847398160, −12.6053560437, −12.1753106139, −11.6799676714, −10.7792657833, −10.7171639889, −10.1965884440, −9.73707630115, −9.14610703071, −8.89231924120, −8.44426556460, −7.84676406921, −7.45176369658, −7.40493950553, −6.20202908477, −5.63622245063, −5.02699315989, −4.58887381257, −3.89034771754, −3.18386250650, −2.58659110698, −1.31411593906, 0, 1.31411593906, 2.58659110698, 3.18386250650, 3.89034771754, 4.58887381257, 5.02699315989, 5.63622245063, 6.20202908477, 7.40493950553, 7.45176369658, 7.84676406921, 8.44426556460, 8.89231924120, 9.14610703071, 9.73707630115, 10.1965884440, 10.7171639889, 10.7792657833, 11.6799676714, 12.1753106139, 12.6053560437, 13.1847398160, 13.3562933455, 13.6791203364, 14.3182182802

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