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  − 2.46·2-s + 4.08·4-s + 3.46·5-s − 7-s − 5.14·8-s − 8.55·10-s − 0.653·13-s + 2.46·14-s + 4.51·16-s + 1.13·17-s − 6.07·19-s + 14.1·20-s + 6.66·23-s + 7.01·25-s + 1.61·26-s − 4.08·28-s + 4.57·29-s + 2.79·31-s − 0.854·32-s − 2.80·34-s − 3.46·35-s − 0.439·37-s + 14.9·38-s − 17.8·40-s + 5.90·41-s + 8.70·43-s − 16.4·46-s + ⋯
L(s)  = 1  − 1.74·2-s + 2.04·4-s + 1.55·5-s − 0.377·7-s − 1.81·8-s − 2.70·10-s − 0.181·13-s + 0.659·14-s + 1.12·16-s + 0.275·17-s − 1.39·19-s + 3.16·20-s + 1.39·23-s + 1.40·25-s + 0.316·26-s − 0.771·28-s + 0.849·29-s + 0.502·31-s − 0.150·32-s − 0.481·34-s − 0.585·35-s − 0.0722·37-s + 2.43·38-s − 2.81·40-s + 0.921·41-s + 1.32·43-s − 2.42·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.156041921$
$L(\frac12)$  $\approx$  $1.156041921$
$L(\frac{3}{2})$   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 \( 1 + 2.46T + 2T^{2} \)
5 \( 1 - 3.46T + 5T^{2} \)
13 \( 1 + 0.653T + 13T^{2} \)
17 \( 1 - 1.13T + 17T^{2} \)
19 \( 1 + 6.07T + 19T^{2} \)
23 \( 1 - 6.66T + 23T^{2} \)
29 \( 1 - 4.57T + 29T^{2} \)
31 \( 1 - 2.79T + 31T^{2} \)
37 \( 1 + 0.439T + 37T^{2} \)
41 \( 1 - 5.90T + 41T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 - 0.604T + 47T^{2} \)
53 \( 1 + 9.82T + 53T^{2} \)
59 \( 1 + 1.69T + 59T^{2} \)
61 \( 1 - 6.85T + 61T^{2} \)
67 \( 1 + 6.17T + 67T^{2} \)
71 \( 1 - 5.41T + 71T^{2} \)
73 \( 1 - 6.70T + 73T^{2} \)
79 \( 1 - 2.65T + 79T^{2} \)
83 \( 1 + 6.69T + 83T^{2} \)
89 \( 1 - 0.698T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−8.099908736678348713686109171774, −7.22166383849783915466312319030, −6.57168645145173086323822306483, −6.18907506498326814303275533168, −5.35992946914113371590051322724, −4.37995559724002057813364698914, −2.91039452501965914964152793777, −2.42360475290526646485775332060, −1.57633295442286144064261307414, −0.72496950403575526330106149715, 0.72496950403575526330106149715, 1.57633295442286144064261307414, 2.42360475290526646485775332060, 2.91039452501965914964152793777, 4.37995559724002057813364698914, 5.35992946914113371590051322724, 6.18907506498326814303275533168, 6.57168645145173086323822306483, 7.22166383849783915466312319030, 8.099908736678348713686109171774

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