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  − 1.31·2-s − 0.276·4-s + 3.45·5-s − 7-s + 2.98·8-s − 4.54·10-s + 5.53·13-s + 1.31·14-s − 3.37·16-s + 4.96·17-s − 8.22·19-s − 0.955·20-s + 7.56·23-s + 6.96·25-s − 7.27·26-s + 0.276·28-s − 1.47·29-s + 2.41·31-s − 1.55·32-s − 6.51·34-s − 3.45·35-s − 2.83·37-s + 10.8·38-s + 10.3·40-s + 6.93·41-s − 5.62·43-s − 9.92·46-s + ⋯
L(s)  = 1  − 0.928·2-s − 0.138·4-s + 1.54·5-s − 0.377·7-s + 1.05·8-s − 1.43·10-s + 1.53·13-s + 0.350·14-s − 0.842·16-s + 1.20·17-s − 1.88·19-s − 0.213·20-s + 1.57·23-s + 1.39·25-s − 1.42·26-s + 0.0522·28-s − 0.272·29-s + 0.433·31-s − 0.274·32-s − 1.11·34-s − 0.584·35-s − 0.466·37-s + 1.75·38-s + 1.63·40-s + 1.08·41-s − 0.858·43-s − 1.46·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.721479205$
$L(\frac12)$  $\approx$  $1.721479205$
$L(\frac{3}{2})$   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 \( 1 + 1.31T + 2T^{2} \)
5 \( 1 - 3.45T + 5T^{2} \)
13 \( 1 - 5.53T + 13T^{2} \)
17 \( 1 - 4.96T + 17T^{2} \)
19 \( 1 + 8.22T + 19T^{2} \)
23 \( 1 - 7.56T + 23T^{2} \)
29 \( 1 + 1.47T + 29T^{2} \)
31 \( 1 - 2.41T + 31T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
41 \( 1 - 6.93T + 41T^{2} \)
43 \( 1 + 5.62T + 43T^{2} \)
47 \( 1 - 0.0417T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 - 2.99T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 9.95T + 67T^{2} \)
71 \( 1 - 6.99T + 71T^{2} \)
73 \( 1 + 7.38T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 9.64T + 83T^{2} \)
89 \( 1 - 7.94T + 89T^{2} \)
97 \( 1 + 2.35T + 97T^{2} \)
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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

−8.228962149156793701493495684684, −7.12032329512673300201763580721, −6.55682427552178822566661954847, −5.84279717311294733397560525016, −5.29005199363465800925661472042, −4.32134000451620516796221250343, −3.45356032442340594130384944554, −2.40345236211070671232720117055, −1.53633250786120851048187376996, −0.833526768094489858765390349809, 0.833526768094489858765390349809, 1.53633250786120851048187376996, 2.40345236211070671232720117055, 3.45356032442340594130384944554, 4.32134000451620516796221250343, 5.29005199363465800925661472042, 5.84279717311294733397560525016, 6.55682427552178822566661954847, 7.12032329512673300201763580721, 8.228962149156793701493495684684

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