Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7 \cdot 13 \cdot 41 $
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 + 4-s − 3·5-s + 6-s + 7-s + 8-s + 9-s − 3·10-s − 3·11-s + 12-s + 13-s + 14-s − 3·15-s + 16-s + 3·17-s + 18-s − 4·19-s − 3·20-s + 21-s − 3·22-s + 3·23-s + 24-s + 4·25-s + 26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.917·19-s − 0.670·20-s + 0.218·21-s − 0.639·22-s + 0.625·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(22386\)    =    \(2 \cdot 3 \cdot 7 \cdot 13 \cdot 41\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{22386} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 22386,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;13,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;13,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
41 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−15.46200886233383, −15.26092898931847, −14.82418737512342, −14.25042785881127, −13.57244735021068, −13.09124899375459, −12.60958922868978, −12.00338206558609, −11.54513029080489, −10.90317203863510, −10.52071597617804, −9.799799433397628, −8.935531367205291, −8.270309870803123, −8.048210620904663, −7.253395059805809, −7.035368302725261, −5.985508058632391, −5.320674912661955, −4.736411423125310, −3.947333129131250, −3.679141932781375, −2.852620997518852, −2.203647704866938, −1.180034236407020, 0, 1.180034236407020, 2.203647704866938, 2.852620997518852, 3.679141932781375, 3.947333129131250, 4.736411423125310, 5.320674912661955, 5.985508058632391, 7.035368302725261, 7.253395059805809, 8.048210620904663, 8.270309870803123, 8.935531367205291, 9.799799433397628, 10.52071597617804, 10.90317203863510, 11.54513029080489, 12.00338206558609, 12.60958922868978, 13.09124899375459, 13.57244735021068, 14.25042785881127, 14.82418737512342, 15.26092898931847, 15.46200886233383

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