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 + 0.769·3-s + 4-s + 5-s + 0.769·6-s + 4.71·7-s + 8-s − 2.40·9-s + 10-s − 11-s + 0.769·12-s + 4.82·13-s + 4.71·14-s + 0.769·15-s + 16-s + 6.87·17-s − 2.40·18-s − 6.08·19-s + 20-s + 3.63·21-s − 22-s − 2.82·23-s + 0.769·24-s + 25-s + 4.82·26-s − 4.16·27-s + 4.71·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.444·3-s + 0.5·4-s + 0.447·5-s + 0.314·6-s + 1.78·7-s + 0.353·8-s − 0.802·9-s + 0.316·10-s − 0.301·11-s + 0.222·12-s + 1.33·13-s + 1.26·14-s + 0.198·15-s + 0.250·16-s + 1.66·17-s − 0.567·18-s − 1.39·19-s + 0.223·20-s + 0.792·21-s − 0.213·22-s − 0.588·23-s + 0.157·24-s + 0.200·25-s + 0.945·26-s − 0.801·27-s + 0.891·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.062046442$
$L(\frac12)$  $\approx$  $5.062046442$
$L(\frac{3}{2})$   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 \( 1 - 0.769T + 3T^{2} \)
7 \( 1 - 4.71T + 7T^{2} \)
13 \( 1 - 4.82T + 13T^{2} \)
17 \( 1 - 6.87T + 17T^{2} \)
19 \( 1 + 6.08T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 0.648T + 29T^{2} \)
31 \( 1 - 8.04T + 31T^{2} \)
37 \( 1 - 1.84T + 37T^{2} \)
41 \( 1 + 5.87T + 41T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 - 8.05T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 2.87T + 67T^{2} \)
71 \( 1 - 2.63T + 71T^{2} \)
73 \( 1 - 1.83T + 73T^{2} \)
79 \( 1 + 9.14T + 79T^{2} \)
83 \( 1 + 2.49T + 83T^{2} \)
89 \( 1 + 6.18T + 89T^{2} \)
97 \( 1 - 8.98T + 97T^{2} \)
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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

−8.256531320513500315645994170410, −7.87640308291083026341630777682, −6.67899183684894855714161799685, −5.89142790914740921104454451816, −5.39822046018795609631869413902, −4.60330164821910565180839625323, −3.80327784303591440449470092468, −2.90125705657675530530840176826, −2.00828835588433119446512108059, −1.23439166033214821112857177611, 1.23439166033214821112857177611, 2.00828835588433119446512108059, 2.90125705657675530530840176826, 3.80327784303591440449470092468, 4.60330164821910565180839625323, 5.39822046018795609631869413902, 5.89142790914740921104454451816, 6.67899183684894855714161799685, 7.87640308291083026341630777682, 8.256531320513500315645994170410

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