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 − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s + 11-s − 3·12-s + 14-s − 3·15-s + 16-s − 3·17-s + 6·18-s − 19-s + 20-s − 3·21-s + 22-s − 4·23-s − 3·24-s + 25-s − 9·27-s + 28-s − 5·29-s − 3·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 0.301·11-s − 0.866·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 0.229·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 1.73·27-s + 0.188·28-s − 0.928·29-s − 0.547·30-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  :  $\chi_{4730} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 - T \)
43 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−17.84675313983433, −17.37144243692654, −17.12728629931520, −16.17314131172689, −15.95467970084369, −15.16654536832013, −14.41239237572319, −13.81392100269725, −13.00048065709477, −12.57794938161210, −11.94069544365215, −11.29917364657803, −10.94953670876546, −10.25506790952758, −9.592626527551025, −8.622665179781464, −7.613742688823653, −6.747634030335561, −6.464231475252572, −5.465713684870454, −5.306018404096157, −4.354115097724995, −3.718242750676210, −2.207203146962727, −1.413343267926647, 0, 1.413343267926647, 2.207203146962727, 3.718242750676210, 4.354115097724995, 5.306018404096157, 5.465713684870454, 6.464231475252572, 6.747634030335561, 7.613742688823653, 8.622665179781464, 9.592626527551025, 10.25506790952758, 10.94953670876546, 11.29917364657803, 11.94069544365215, 12.57794938161210, 13.00048065709477, 13.81392100269725, 14.41239237572319, 15.16654536832013, 15.95467970084369, 16.17314131172689, 17.12728629931520, 17.37144243692654, 17.84675313983433

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