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 − 2.17·3-s + 4-s − 5-s + 2.17·6-s + 4.63·7-s − 8-s + 1.74·9-s + 10-s − 11-s − 2.17·12-s − 5.80·13-s − 4.63·14-s + 2.17·15-s + 16-s − 7.18·17-s − 1.74·18-s − 6.16·19-s − 20-s − 10.1·21-s + 22-s − 7.80·23-s + 2.17·24-s + 25-s + 5.80·26-s + 2.73·27-s + 4.63·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.25·3-s + 0.5·4-s − 0.447·5-s + 0.889·6-s + 1.75·7-s − 0.353·8-s + 0.581·9-s + 0.316·10-s − 0.301·11-s − 0.628·12-s − 1.61·13-s − 1.23·14-s + 0.562·15-s + 0.250·16-s − 1.74·17-s − 0.410·18-s − 1.41·19-s − 0.223·20-s − 2.20·21-s + 0.213·22-s − 1.62·23-s + 0.444·24-s + 0.200·25-s + 1.13·26-s + 0.526·27-s + 0.876·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2935560370$
$L(\frac12)$  $\approx$  $0.2935560370$
$L(\frac{3}{2})$   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 \( 1 + 2.17T + 3T^{2} \)
7 \( 1 - 4.63T + 7T^{2} \)
13 \( 1 + 5.80T + 13T^{2} \)
17 \( 1 + 7.18T + 17T^{2} \)
19 \( 1 + 6.16T + 19T^{2} \)
23 \( 1 + 7.80T + 23T^{2} \)
29 \( 1 - 1.51T + 29T^{2} \)
31 \( 1 + 3.25T + 31T^{2} \)
37 \( 1 - 3.52T + 37T^{2} \)
41 \( 1 + 8.02T + 41T^{2} \)
47 \( 1 + 7.28T + 47T^{2} \)
53 \( 1 + 0.333T + 53T^{2} \)
59 \( 1 + 1.57T + 59T^{2} \)
61 \( 1 - 14.7T + 61T^{2} \)
67 \( 1 - 3.48T + 67T^{2} \)
71 \( 1 + 9.42T + 71T^{2} \)
73 \( 1 + 3.01T + 73T^{2} \)
79 \( 1 - 5.06T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−8.275069460326867559758662315923, −7.65224012761748102797103866418, −6.87396627553210055071140944227, −6.25147541859934528193828411027, −5.24514132889086730512075014962, −4.73981710531103466825100125279, −4.14836510832555559044771840364, −2.36266546166539410307706445465, −1.86845399323764907699654322076, −0.33903209887734812572239508407, 0.33903209887734812572239508407, 1.86845399323764907699654322076, 2.36266546166539410307706445465, 4.14836510832555559044771840364, 4.73981710531103466825100125279, 5.24514132889086730512075014962, 6.25147541859934528193828411027, 6.87396627553210055071140944227, 7.65224012761748102797103866418, 8.275069460326867559758662315923

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