Properties

Degree 4
Conductor $ 2 \cdot 3^{4} \cdot 619 $
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 − 2·3-s + 4-s + 5-s − 2·6-s − 5·7-s − 8-s + 9-s + 10-s + 2·11-s − 2·12-s − 5·14-s − 2·15-s − 3·16-s + 9·17-s + 18-s − 3·19-s + 20-s + 10·21-s + 2·22-s + 3·23-s + 2·24-s + 4·27-s − 5·28-s + 29-s − 2·30-s − 3·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.577·12-s − 1.33·14-s − 0.516·15-s − 3/4·16-s + 2.18·17-s + 0.235·18-s − 0.688·19-s + 0.223·20-s + 2.18·21-s + 0.426·22-s + 0.625·23-s + 0.408·24-s + 0.769·27-s − 0.944·28-s + 0.185·29-s − 0.365·30-s − 0.538·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100278 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100278 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100278\)    =    \(2 \cdot 3^{4} \cdot 619\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100278} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 100278,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;619\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;619\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - p T + p T^{2} ) \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
619$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good5$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 11 T + 96 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T + 68 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 7 T + 119 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 20 T + 220 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 17 T + 151 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
97$D_{4}$ \( 1 - 5 T + 94 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
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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

−14.2468095405, −13.8162565334, −13.2090723980, −12.8604206324, −12.5513878046, −12.0863881316, −11.8348314058, −11.3754971569, −10.7239681852, −10.2675008843, −9.93549699937, −9.46696495946, −8.99644979286, −8.43491858394, −7.56665409835, −7.01182564085, −6.44135007875, −6.21107232654, −5.93572527962, −5.16614375569, −4.85723841108, −3.72477157838, −3.29477280503, −2.86364856634, −1.50358718987, 0, 1.50358718987, 2.86364856634, 3.29477280503, 3.72477157838, 4.85723841108, 5.16614375569, 5.93572527962, 6.21107232654, 6.44135007875, 7.01182564085, 7.56665409835, 8.43491858394, 8.99644979286, 9.46696495946, 9.93549699937, 10.2675008843, 10.7239681852, 11.3754971569, 11.8348314058, 12.0863881316, 12.5513878046, 12.8604206324, 13.2090723980, 13.8162565334, 14.2468095405

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