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 − 1.30·3-s + 4-s − 5-s − 1.30·6-s + 3.42·7-s + 8-s − 1.30·9-s − 10-s + 11-s − 1.30·12-s + 2.16·13-s + 3.42·14-s + 1.30·15-s + 16-s + 6.19·17-s − 1.30·18-s + 4.97·19-s − 20-s − 4.45·21-s + 22-s − 4.95·23-s − 1.30·24-s + 25-s + 2.16·26-s + 5.60·27-s + 3.42·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.750·3-s + 0.5·4-s − 0.447·5-s − 0.530·6-s + 1.29·7-s + 0.353·8-s − 0.436·9-s − 0.316·10-s + 0.301·11-s − 0.375·12-s + 0.600·13-s + 0.914·14-s + 0.335·15-s + 0.250·16-s + 1.50·17-s − 0.308·18-s + 1.14·19-s − 0.223·20-s − 0.971·21-s + 0.213·22-s − 1.03·23-s − 0.265·24-s + 0.200·25-s + 0.424·26-s + 1.07·27-s + 0.646·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.722612624$
$L(\frac12)$  $\approx$  $2.722612624$
$L(\frac{3}{2})$   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 \( 1 + 1.30T + 3T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
13 \( 1 - 2.16T + 13T^{2} \)
17 \( 1 - 6.19T + 17T^{2} \)
19 \( 1 - 4.97T + 19T^{2} \)
23 \( 1 + 4.95T + 23T^{2} \)
29 \( 1 + 8.41T + 29T^{2} \)
31 \( 1 - 7.34T + 31T^{2} \)
37 \( 1 + 4.92T + 37T^{2} \)
41 \( 1 + 6.22T + 41T^{2} \)
47 \( 1 - 4.34T + 47T^{2} \)
53 \( 1 - 0.623T + 53T^{2} \)
59 \( 1 - 2.83T + 59T^{2} \)
61 \( 1 + 4.29T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 + 5.78T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 9.34T + 79T^{2} \)
83 \( 1 + 7.05T + 83T^{2} \)
89 \( 1 - 8.33T + 89T^{2} \)
97 \( 1 - 2.72T + 97T^{2} \)
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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

−8.045540622189759441001704386403, −7.63118030441265773269644193828, −6.72530377736862064415072705874, −5.77576694330399025963049065783, −5.44030746898904651314554908894, −4.73307480329057718312679545504, −3.82718391231220037556830403442, −3.15021362783406321183570349801, −1.83235870003294553176080502906, −0.903086152313642568672768765287, 0.903086152313642568672768765287, 1.83235870003294553176080502906, 3.15021362783406321183570349801, 3.82718391231220037556830403442, 4.73307480329057718312679545504, 5.44030746898904651314554908894, 5.77576694330399025963049065783, 6.72530377736862064415072705874, 7.63118030441265773269644193828, 8.045540622189759441001704386403

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