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 − 3.45·3-s + 4-s − 5-s − 3.45·6-s − 3.54·7-s + 8-s + 8.96·9-s − 10-s + 11-s − 3.45·12-s + 5.54·13-s − 3.54·14-s + 3.45·15-s + 16-s − 6.38·17-s + 8.96·18-s + 7.62·19-s − 20-s + 12.2·21-s + 22-s − 4.31·23-s − 3.45·24-s + 25-s + 5.54·26-s − 20.6·27-s − 3.54·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.99·3-s + 0.5·4-s − 0.447·5-s − 1.41·6-s − 1.33·7-s + 0.353·8-s + 2.98·9-s − 0.316·10-s + 0.301·11-s − 0.998·12-s + 1.53·13-s − 0.947·14-s + 0.893·15-s + 0.250·16-s − 1.54·17-s + 2.11·18-s + 1.74·19-s − 0.223·20-s + 2.67·21-s + 0.213·22-s − 0.899·23-s − 0.706·24-s + 0.200·25-s + 1.08·26-s − 3.96·27-s − 0.669·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.013402037$
$L(\frac12)$  $\approx$  $1.013402037$
$L(\frac{3}{2})$   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 \( 1 + 3.45T + 3T^{2} \)
7 \( 1 + 3.54T + 7T^{2} \)
13 \( 1 - 5.54T + 13T^{2} \)
17 \( 1 + 6.38T + 17T^{2} \)
19 \( 1 - 7.62T + 19T^{2} \)
23 \( 1 + 4.31T + 23T^{2} \)
29 \( 1 - 2.64T + 29T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 + 2.16T + 37T^{2} \)
41 \( 1 + 3.56T + 41T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 - 2.96T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 0.437T + 61T^{2} \)
67 \( 1 - 6.14T + 67T^{2} \)
71 \( 1 + 5.21T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−8.037855262721241007713536885556, −7.03223558164473834958196006026, −6.55643771395487621907754261387, −6.18969826771701615039899396710, −5.45840277092474693422281317593, −4.68218760073984666669489083123, −3.89744275825743708982498058913, −3.32906951214174325225585066429, −1.66232235589244256589799960761, −0.56726266658005982415677412700, 0.56726266658005982415677412700, 1.66232235589244256589799960761, 3.32906951214174325225585066429, 3.89744275825743708982498058913, 4.68218760073984666669489083123, 5.45840277092474693422281317593, 6.18969826771701615039899396710, 6.55643771395487621907754261387, 7.03223558164473834958196006026, 8.037855262721241007713536885556

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