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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 4·17-s + 18-s − 8·19-s − 20-s + 21-s − 2·22-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 − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-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  =  0
Selberg data  =  $(2,\ 22386,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.026544157$
$L(\frac12)$  $\approx$  $4.026544157$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 + T \)
41 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.21613226555452, −15.08837240625950, −14.45671219933780, −13.90947794026592, −13.41667323463503, −12.80193182785214, −12.34214713713175, −11.90664579686695, −11.03307330206842, −10.79760235043915, −9.984686321799649, −9.572548153122640, −8.563516753023284, −8.153559918028390, −7.797153306108810, −6.964357585333038, −6.513418048751630, −5.639117914361068, −5.085018983778566, −4.359143687369953, −3.915097926303344, −3.117827134453124, −2.459441845580925, −1.823530723223458, −0.6853363922700653, 0.6853363922700653, 1.823530723223458, 2.459441845580925, 3.117827134453124, 3.915097926303344, 4.359143687369953, 5.085018983778566, 5.639117914361068, 6.513418048751630, 6.964357585333038, 7.797153306108810, 8.153559918028390, 8.563516753023284, 9.572548153122640, 9.984686321799649, 10.79760235043915, 11.03307330206842, 11.90664579686695, 12.34214713713175, 12.80193182785214, 13.41667323463503, 13.90947794026592, 14.45671219933780, 15.08837240625950, 15.21613226555452

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