Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.228·3-s + 4-s + 5-s + 0.228·6-s − 2.81·7-s − 8-s − 2.94·9-s − 10-s + 11-s − 0.228·12-s + 1.12·13-s + 2.81·14-s − 0.228·15-s + 16-s − 1.66·17-s + 2.94·18-s + 7.42·19-s + 20-s + 0.642·21-s − 22-s − 0.875·23-s + 0.228·24-s + 25-s − 1.12·26-s + 1.35·27-s − 2.81·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.131·3-s + 0.5·4-s + 0.447·5-s + 0.0931·6-s − 1.06·7-s − 0.353·8-s − 0.982·9-s − 0.316·10-s + 0.301·11-s − 0.0658·12-s + 0.311·13-s + 0.753·14-s − 0.0588·15-s + 0.250·16-s − 0.403·17-s + 0.694·18-s + 1.70·19-s + 0.223·20-s + 0.140·21-s − 0.213·22-s − 0.182·23-s + 0.0465·24-s + 0.200·25-s − 0.220·26-s + 0.261·27-s − 0.532·28-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  =  1
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 0.228T + 3T^{2} \)
7 \( 1 + 2.81T + 7T^{2} \)
13 \( 1 - 1.12T + 13T^{2} \)
17 \( 1 + 1.66T + 17T^{2} \)
19 \( 1 - 7.42T + 19T^{2} \)
23 \( 1 + 0.875T + 23T^{2} \)
29 \( 1 + 4.72T + 29T^{2} \)
31 \( 1 + 2.66T + 31T^{2} \)
37 \( 1 + 5.63T + 37T^{2} \)
41 \( 1 - 7.80T + 41T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 + 3.40T + 53T^{2} \)
59 \( 1 - 0.759T + 59T^{2} \)
61 \( 1 - 4.85T + 61T^{2} \)
67 \( 1 + 2.24T + 67T^{2} \)
71 \( 1 - 7.41T + 71T^{2} \)
73 \( 1 - 2.51T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 + 6.50T + 97T^{2} \)
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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

−7.966568674040201317700731593654, −7.18773142630946122749843759958, −6.56606213221029761755497252708, −5.74671581251072604499444093161, −5.40390475892420037672345549994, −3.94187274851885488736289236033, −3.15304496131833790655900421742, −2.42421035640832977565741737549, −1.19532708609762698666342566592, 0, 1.19532708609762698666342566592, 2.42421035640832977565741737549, 3.15304496131833790655900421742, 3.94187274851885488736289236033, 5.40390475892420037672345549994, 5.74671581251072604499444093161, 6.56606213221029761755497252708, 7.18773142630946122749843759958, 7.966568674040201317700731593654

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