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 − 3.42·3-s + 4-s + 5-s + 3.42·6-s − 2.60·7-s − 8-s + 8.73·9-s − 10-s + 11-s − 3.42·12-s + 2.63·13-s + 2.60·14-s − 3.42·15-s + 16-s + 0.262·17-s − 8.73·18-s − 1.75·19-s + 20-s + 8.91·21-s − 22-s + 0.630·23-s + 3.42·24-s + 25-s − 2.63·26-s − 19.6·27-s − 2.60·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.97·3-s + 0.5·4-s + 0.447·5-s + 1.39·6-s − 0.983·7-s − 0.353·8-s + 2.91·9-s − 0.316·10-s + 0.301·11-s − 0.988·12-s + 0.729·13-s + 0.695·14-s − 0.884·15-s + 0.250·16-s + 0.0637·17-s − 2.05·18-s − 0.402·19-s + 0.223·20-s + 1.94·21-s − 0.213·22-s + 0.131·23-s + 0.699·24-s + 0.200·25-s − 0.515·26-s − 3.78·27-s − 0.491·28-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 \( 1 + 3.42T + 3T^{2} \)
7 \( 1 + 2.60T + 7T^{2} \)
13 \( 1 - 2.63T + 13T^{2} \)
17 \( 1 - 0.262T + 17T^{2} \)
19 \( 1 + 1.75T + 19T^{2} \)
23 \( 1 - 0.630T + 23T^{2} \)
29 \( 1 + 7.05T + 29T^{2} \)
31 \( 1 - 1.60T + 31T^{2} \)
37 \( 1 + 2.81T + 37T^{2} \)
41 \( 1 - 7.41T + 41T^{2} \)
47 \( 1 + 6.45T + 47T^{2} \)
53 \( 1 - 5.51T + 53T^{2} \)
59 \( 1 - 3.57T + 59T^{2} \)
61 \( 1 - 0.307T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 8.53T + 71T^{2} \)
73 \( 1 - 2.33T + 73T^{2} \)
79 \( 1 - 8.43T + 79T^{2} \)
83 \( 1 + 3.47T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 - 2.91T + 97T^{2} \)
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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

−7.68988610309357651313574373680, −6.98728974812065412701299959501, −6.35628755940033833790535372081, −5.99861957958728479666895086718, −5.30206250758380988193489854510, −4.30887620960043381200187641779, −3.43120659158157333485785253056, −1.94640953834633423784926627253, −1.01342194243731750262531311886, 0, 1.01342194243731750262531311886, 1.94640953834633423784926627253, 3.43120659158157333485785253056, 4.30887620960043381200187641779, 5.30206250758380988193489854510, 5.99861957958728479666895086718, 6.35628755940033833790535372081, 6.98728974812065412701299959501, 7.68988610309357651313574373680

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