Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 5-s − 7-s − 2·10-s + 4·13-s + 2·14-s − 4·16-s + 17-s + 2·20-s − 4·23-s − 4·25-s − 8·26-s − 2·28-s − 2·31-s + 8·32-s − 2·34-s − 35-s + 6·37-s − 2·41-s + 3·43-s + 8·46-s − 7·47-s + 49-s + 8·50-s + 8·52-s − 12·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 0.632·10-s + 1.10·13-s + 0.534·14-s − 16-s + 0.242·17-s + 0.447·20-s − 0.834·23-s − 4/5·25-s − 1.56·26-s − 0.377·28-s − 0.359·31-s + 1.41·32-s − 0.342·34-s − 0.169·35-s + 0.986·37-s − 0.312·41-s + 0.457·43-s + 1.17·46-s − 1.02·47-s + 1/7·49-s + 1.13·50-s + 1.10·52-s − 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8918462013$
$L(\frac12)$  $\approx$  $0.8918462013$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.135074811419118747409365767925, −7.39867996347574951100690083975, −6.53793141367775547122729367257, −6.10545440412883978115117003557, −5.23781087715456662834297829318, −4.18797630475859956172217899884, −3.43572668009949702378896514059, −2.30088973459600653765576051863, −1.58608166618136765220416168978, −0.60643767119777141246349274535, 0.60643767119777141246349274535, 1.58608166618136765220416168978, 2.30088973459600653765576051863, 3.43572668009949702378896514059, 4.18797630475859956172217899884, 5.23781087715456662834297829318, 6.10545440412883978115117003557, 6.53793141367775547122729367257, 7.39867996347574951100690083975, 8.135074811419118747409365767925

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