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 + 0.476·3-s + 4-s + 5-s − 0.476·6-s + 2.32·7-s − 8-s − 2.77·9-s − 10-s + 11-s + 0.476·12-s + 7.04·13-s − 2.32·14-s + 0.476·15-s + 16-s + 2.62·17-s + 2.77·18-s + 2.34·19-s + 20-s + 1.10·21-s − 22-s − 1.69·23-s − 0.476·24-s + 25-s − 7.04·26-s − 2.75·27-s + 2.32·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.275·3-s + 0.5·4-s + 0.447·5-s − 0.194·6-s + 0.879·7-s − 0.353·8-s − 0.924·9-s − 0.316·10-s + 0.301·11-s + 0.137·12-s + 1.95·13-s − 0.621·14-s + 0.123·15-s + 0.250·16-s + 0.636·17-s + 0.653·18-s + 0.537·19-s + 0.223·20-s + 0.242·21-s − 0.213·22-s − 0.353·23-s − 0.0973·24-s + 0.200·25-s − 1.38·26-s − 0.529·27-s + 0.439·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.150237765$
$L(\frac12)$  $\approx$  $2.150237765$
$L(\frac{3}{2})$   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 \( 1 - 0.476T + 3T^{2} \)
7 \( 1 - 2.32T + 7T^{2} \)
13 \( 1 - 7.04T + 13T^{2} \)
17 \( 1 - 2.62T + 17T^{2} \)
19 \( 1 - 2.34T + 19T^{2} \)
23 \( 1 + 1.69T + 23T^{2} \)
29 \( 1 - 7.70T + 29T^{2} \)
31 \( 1 - 1.97T + 31T^{2} \)
37 \( 1 + 3.78T + 37T^{2} \)
41 \( 1 - 6.55T + 41T^{2} \)
47 \( 1 - 0.404T + 47T^{2} \)
53 \( 1 + 9.69T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 + 6.27T + 61T^{2} \)
67 \( 1 - 0.865T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 7.72T + 73T^{2} \)
79 \( 1 - 4.49T + 79T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−8.336311666297893924208565308669, −7.931829396210317815065884873452, −6.90417770515121470508654023836, −6.04034986237231516966977746335, −5.67688860606107276709726547545, −4.59293430765698521378314976703, −3.53115878793252867028537792968, −2.80605946817563626912583530863, −1.68016166398938486439477754057, −0.975291240593934978847746102773, 0.975291240593934978847746102773, 1.68016166398938486439477754057, 2.80605946817563626912583530863, 3.53115878793252867028537792968, 4.59293430765698521378314976703, 5.67688860606107276709726547545, 6.04034986237231516966977746335, 6.90417770515121470508654023836, 7.931829396210317815065884873452, 8.336311666297893924208565308669

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