Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
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 − 0.580·3-s + 4-s − 5-s + 0.580·6-s + 1.64·7-s − 8-s − 2.66·9-s + 10-s − 11-s − 0.580·12-s + 2.95·13-s − 1.64·14-s + 0.580·15-s + 16-s + 1.81·17-s + 2.66·18-s − 3.80·19-s − 20-s − 0.955·21-s + 22-s + 0.952·23-s + 0.580·24-s + 25-s − 2.95·26-s + 3.28·27-s + 1.64·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.335·3-s + 0.5·4-s − 0.447·5-s + 0.236·6-s + 0.622·7-s − 0.353·8-s − 0.887·9-s + 0.316·10-s − 0.301·11-s − 0.167·12-s + 0.819·13-s − 0.440·14-s + 0.149·15-s + 0.250·16-s + 0.439·17-s + 0.627·18-s − 0.872·19-s − 0.223·20-s − 0.208·21-s + 0.213·22-s + 0.198·23-s + 0.118·24-s + 0.200·25-s − 0.579·26-s + 0.632·27-s + 0.311·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4730\)    =    \(2 \cdot 5 \cdot 11 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4730} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9228729978$
$L(\frac12)$  $\approx$  $0.9228729978$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 + T \)
43 \( 1 - T \)
good3 \( 1 + 0.580T + 3T^{2} \)
7 \( 1 - 1.64T + 7T^{2} \)
13 \( 1 - 2.95T + 13T^{2} \)
17 \( 1 - 1.81T + 17T^{2} \)
19 \( 1 + 3.80T + 19T^{2} \)
23 \( 1 - 0.952T + 23T^{2} \)
29 \( 1 - 3.68T + 29T^{2} \)
31 \( 1 - 3.41T + 31T^{2} \)
37 \( 1 + 2.19T + 37T^{2} \)
41 \( 1 + 5.45T + 41T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 8.91T + 53T^{2} \)
59 \( 1 + 3.10T + 59T^{2} \)
61 \( 1 + 9.87T + 61T^{2} \)
67 \( 1 - 6.18T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 0.229T + 73T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 - 2.72T + 83T^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 + 3.05T + 97T^{2} \)
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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

−8.216777617208611816398043149411, −7.947753930735778410277641217083, −6.78324054003905559582675586973, −6.30957583919492958568095498685, −5.39311780064355301143995952710, −4.72224269886000119621259658903, −3.62781019133375249072988885706, −2.81253985869006210495157234959, −1.72531420680418500724607370469, −0.60428133488803621536962010390, 0.60428133488803621536962010390, 1.72531420680418500724607370469, 2.81253985869006210495157234959, 3.62781019133375249072988885706, 4.72224269886000119621259658903, 5.39311780064355301143995952710, 6.30957583919492958568095498685, 6.78324054003905559582675586973, 7.947753930735778410277641217083, 8.216777617208611816398043149411

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