Properties

Degree 4
Conductor $ 2 \cdot 11^{2} \cdot 19^{2} $
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 + 4-s + 5-s + 8-s + 9-s − 10-s − 3·11-s − 8·13-s − 3·16-s − 18-s − 4·19-s + 20-s + 3·22-s + 2·23-s − 3·25-s + 8·26-s − 2·29-s + 5·32-s + 36-s + 4·38-s + 40-s − 10·41-s − 3·44-s + 45-s − 2·46-s + 4·47-s − 11·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 2.21·13-s − 3/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.639·22-s + 0.417·23-s − 3/5·25-s + 1.56·26-s − 0.371·29-s + 0.883·32-s + 1/6·36-s + 0.648·38-s + 0.158·40-s − 1.56·41-s − 0.452·44-s + 0.149·45-s − 0.294·46-s + 0.583·47-s − 1.57·49-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(87362\)    =    \(2 \cdot 11^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{87362} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 87362,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;19\}$, 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$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
11$C_2$ \( 1 + 3 T + p T^{2} \)
19$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 27 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 71 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 113 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 181 T^{2} + p^{2} T^{4} \)
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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

−9.450223779291344770807595463754, −9.140781653340821274478576792169, −8.246617768836988210813697607109, −7.996925043161845275175381485328, −7.47472357059831168214977383824, −6.91320625381387899247441931584, −6.66737315612773205927859990214, −5.77349495078356174773613983416, −5.09725966565904174656575072428, −4.79862490565530800982393900729, −4.07549449166981345454199458144, −2.97385313027726012133994344346, −2.31538745683713410959810010695, −1.75487743553433853867216320915, 0, 1.75487743553433853867216320915, 2.31538745683713410959810010695, 2.97385313027726012133994344346, 4.07549449166981345454199458144, 4.79862490565530800982393900729, 5.09725966565904174656575072428, 5.77349495078356174773613983416, 6.66737315612773205927859990214, 6.91320625381387899247441931584, 7.47472357059831168214977383824, 7.996925043161845275175381485328, 8.246617768836988210813697607109, 9.140781653340821274478576792169, 9.450223779291344770807595463754

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