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 + 2.41·3-s + 4-s + 5-s − 2.41·6-s − 5.00·7-s − 8-s + 2.82·9-s − 10-s + 11-s + 2.41·12-s + 7.13·13-s + 5.00·14-s + 2.41·15-s + 16-s − 2.68·17-s − 2.82·18-s − 3.51·19-s + 20-s − 12.0·21-s − 22-s + 6.05·23-s − 2.41·24-s + 25-s − 7.13·26-s − 0.426·27-s − 5.00·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.39·3-s + 0.5·4-s + 0.447·5-s − 0.985·6-s − 1.89·7-s − 0.353·8-s + 0.941·9-s − 0.316·10-s + 0.301·11-s + 0.696·12-s + 1.97·13-s + 1.33·14-s + 0.623·15-s + 0.250·16-s − 0.650·17-s − 0.665·18-s − 0.806·19-s + 0.223·20-s − 2.63·21-s − 0.213·22-s + 1.26·23-s − 0.492·24-s + 0.200·25-s − 1.39·26-s − 0.0820·27-s − 0.946·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.175188713$
$L(\frac12)$  $\approx$  $2.175188713$
$L(\frac{3}{2})$   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 \( 1 - 2.41T + 3T^{2} \)
7 \( 1 + 5.00T + 7T^{2} \)
13 \( 1 - 7.13T + 13T^{2} \)
17 \( 1 + 2.68T + 17T^{2} \)
19 \( 1 + 3.51T + 19T^{2} \)
23 \( 1 - 6.05T + 23T^{2} \)
29 \( 1 + 7.74T + 29T^{2} \)
31 \( 1 - 7.62T + 31T^{2} \)
37 \( 1 - 3.83T + 37T^{2} \)
41 \( 1 - 3.90T + 41T^{2} \)
47 \( 1 + 0.876T + 47T^{2} \)
53 \( 1 + 1.81T + 53T^{2} \)
59 \( 1 - 1.02T + 59T^{2} \)
61 \( 1 - 1.82T + 61T^{2} \)
67 \( 1 - 8.02T + 67T^{2} \)
71 \( 1 - 8.89T + 71T^{2} \)
73 \( 1 + 1.68T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 7.73T + 83T^{2} \)
89 \( 1 + 9.35T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−8.595486558230325263928424979587, −7.78108580310506296331547324604, −6.74859261327264510948033501296, −6.45526648435261376105651662416, −5.73240224241558701690875318639, −4.11037843713323923831481343690, −3.49080736004663848760545740797, −2.86239983941658287586812503419, −2.05690367716154659941945627559, −0.845090759264541758927565292446, 0.845090759264541758927565292446, 2.05690367716154659941945627559, 2.86239983941658287586812503419, 3.49080736004663848760545740797, 4.11037843713323923831481343690, 5.73240224241558701690875318639, 6.45526648435261376105651662416, 6.74859261327264510948033501296, 7.78108580310506296331547324604, 8.595486558230325263928424979587

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