Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 23^{2} $
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 + 4-s + 2·6-s − 7-s − 8-s + 9-s − 2·12-s − 4·13-s + 14-s + 16-s − 6·17-s − 18-s − 2·19-s + 2·21-s + 2·24-s − 5·25-s + 4·26-s + 4·27-s − 28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 36-s − 2·37-s + 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s + 0.436·21-s + 0.408·24-s − 25-s + 0.784·26-s + 0.769·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7406\)    =    \(2 \cdot 7 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7406} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 7406,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;23\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.63148045921673, −17.03505074411956, −16.82473142086000, −16.03674661456657, −15.61363374741032, −14.87668892132566, −14.34806321847780, −13.26564435475815, −12.91150108554483, −12.09107196346880, −11.69195945127466, −10.95610860285343, −10.71012994651170, −9.755231982219801, −9.410810780383984, −8.623004099517610, −7.814821794761255, −7.137449628894325, −6.482017050934037, −6.020258597819443, −5.157923690295609, −4.552851596681342, −3.514998353089305, −2.447583109993457, −1.663406440564645, 0, 0, 1.663406440564645, 2.447583109993457, 3.514998353089305, 4.552851596681342, 5.157923690295609, 6.020258597819443, 6.482017050934037, 7.137449628894325, 7.814821794761255, 8.623004099517610, 9.410810780383984, 9.755231982219801, 10.71012994651170, 10.95610860285343, 11.69195945127466, 12.09107196346880, 12.91150108554483, 13.26564435475815, 14.34806321847780, 14.87668892132566, 15.61363374741032, 16.03674661456657, 16.82473142086000, 17.03505074411956, 17.63148045921673

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