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 + 1.21·3-s + 4-s + 5-s − 1.21·6-s + 3.59·7-s − 8-s − 1.51·9-s − 10-s + 11-s + 1.21·12-s − 4.48·13-s − 3.59·14-s + 1.21·15-s + 16-s + 4.55·17-s + 1.51·18-s + 7.24·19-s + 20-s + 4.37·21-s − 22-s + 1.56·23-s − 1.21·24-s + 25-s + 4.48·26-s − 5.49·27-s + 3.59·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.702·3-s + 0.5·4-s + 0.447·5-s − 0.496·6-s + 1.35·7-s − 0.353·8-s − 0.506·9-s − 0.316·10-s + 0.301·11-s + 0.351·12-s − 1.24·13-s − 0.961·14-s + 0.314·15-s + 0.250·16-s + 1.10·17-s + 0.358·18-s + 1.66·19-s + 0.223·20-s + 0.955·21-s − 0.213·22-s + 0.325·23-s − 0.248·24-s + 0.200·25-s + 0.879·26-s − 1.05·27-s + 0.679·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.362778326$
$L(\frac12)$  $\approx$  $2.362778326$
$L(\frac{3}{2})$   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 \( 1 - 1.21T + 3T^{2} \)
7 \( 1 - 3.59T + 7T^{2} \)
13 \( 1 + 4.48T + 13T^{2} \)
17 \( 1 - 4.55T + 17T^{2} \)
19 \( 1 - 7.24T + 19T^{2} \)
23 \( 1 - 1.56T + 23T^{2} \)
29 \( 1 + 5.71T + 29T^{2} \)
31 \( 1 - 2.63T + 31T^{2} \)
37 \( 1 - 8.17T + 37T^{2} \)
41 \( 1 - 3.08T + 41T^{2} \)
47 \( 1 - 1.64T + 47T^{2} \)
53 \( 1 + 9.99T + 53T^{2} \)
59 \( 1 + 8.65T + 59T^{2} \)
61 \( 1 - 0.979T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 - 0.337T + 71T^{2} \)
73 \( 1 - 7.13T + 73T^{2} \)
79 \( 1 + 8.23T + 79T^{2} \)
83 \( 1 + 1.93T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 - 6.05T + 97T^{2} \)
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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

−8.151373473661472053031881786671, −7.69813476117280161930623898812, −7.34283117663033340651750212786, −6.08666945442756550647617893019, −5.36518568923490436484785341963, −4.74393120268198902481664633734, −3.46433781750023561957123276775, −2.69174817189965820865802188913, −1.88917908675220828248916331041, −0.958065939484708868057225303540, 0.958065939484708868057225303540, 1.88917908675220828248916331041, 2.69174817189965820865802188913, 3.46433781750023561957123276775, 4.74393120268198902481664633734, 5.36518568923490436484785341963, 6.08666945442756550647617893019, 7.34283117663033340651750212786, 7.69813476117280161930623898812, 8.151373473661472053031881786671

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