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 − 3.26·3-s + 4-s − 5-s − 3.26·6-s + 2.73·7-s + 8-s + 7.64·9-s − 10-s + 11-s − 3.26·12-s + 6.11·13-s + 2.73·14-s + 3.26·15-s + 16-s + 7.48·17-s + 7.64·18-s − 7.78·19-s − 20-s − 8.91·21-s + 22-s + 8.88·23-s − 3.26·24-s + 25-s + 6.11·26-s − 15.1·27-s + 2.73·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.88·3-s + 0.5·4-s − 0.447·5-s − 1.33·6-s + 1.03·7-s + 0.353·8-s + 2.54·9-s − 0.316·10-s + 0.301·11-s − 0.941·12-s + 1.69·13-s + 0.730·14-s + 0.842·15-s + 0.250·16-s + 1.81·17-s + 1.80·18-s − 1.78·19-s − 0.223·20-s − 1.94·21-s + 0.213·22-s + 1.85·23-s − 0.665·24-s + 0.200·25-s + 1.19·26-s − 2.91·27-s + 0.516·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.236440121$
$L(\frac12)$  $\approx$  $2.236440121$
$L(\frac{3}{2})$   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 \( 1 + 3.26T + 3T^{2} \)
7 \( 1 - 2.73T + 7T^{2} \)
13 \( 1 - 6.11T + 13T^{2} \)
17 \( 1 - 7.48T + 17T^{2} \)
19 \( 1 + 7.78T + 19T^{2} \)
23 \( 1 - 8.88T + 23T^{2} \)
29 \( 1 - 6.36T + 29T^{2} \)
31 \( 1 - 6.01T + 31T^{2} \)
37 \( 1 + 0.607T + 37T^{2} \)
41 \( 1 - 2.66T + 41T^{2} \)
47 \( 1 - 0.852T + 47T^{2} \)
53 \( 1 + 9.54T + 53T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 - 9.95T + 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 + 4.13T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + 2.52T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 1.91T + 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−8.121151455116124530730966933299, −7.31620368064106851508019399461, −6.49560591002883483264397188158, −6.08827733780176826602407272534, −5.34570957508983747143985002915, −4.62186097166159386515040561096, −4.20847985328712505572730460897, −3.16006526695854062359370712534, −1.45469265277662554040794008604, −0.969721852380763681807924990343, 0.969721852380763681807924990343, 1.45469265277662554040794008604, 3.16006526695854062359370712534, 4.20847985328712505572730460897, 4.62186097166159386515040561096, 5.34570957508983747143985002915, 6.08827733780176826602407272534, 6.49560591002883483264397188158, 7.31620368064106851508019399461, 8.121151455116124530730966933299

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