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 + 2.86·3-s + 4-s − 5-s + 2.86·6-s + 4.06·7-s + 8-s + 5.19·9-s − 10-s + 11-s + 2.86·12-s + 7.12·13-s + 4.06·14-s − 2.86·15-s + 16-s − 7.00·17-s + 5.19·18-s − 2.52·19-s − 20-s + 11.6·21-s + 22-s + 0.631·23-s + 2.86·24-s + 25-s + 7.12·26-s + 6.27·27-s + 4.06·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.65·3-s + 0.5·4-s − 0.447·5-s + 1.16·6-s + 1.53·7-s + 0.353·8-s + 1.73·9-s − 0.316·10-s + 0.301·11-s + 0.826·12-s + 1.97·13-s + 1.08·14-s − 0.739·15-s + 0.250·16-s − 1.69·17-s + 1.22·18-s − 0.580·19-s − 0.223·20-s + 2.53·21-s + 0.213·22-s + 0.131·23-s + 0.584·24-s + 0.200·25-s + 1.39·26-s + 1.20·27-s + 0.768·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.766639031$
$L(\frac12)$  $\approx$  $6.766639031$
$L(\frac{3}{2})$   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 \( 1 - 2.86T + 3T^{2} \)
7 \( 1 - 4.06T + 7T^{2} \)
13 \( 1 - 7.12T + 13T^{2} \)
17 \( 1 + 7.00T + 17T^{2} \)
19 \( 1 + 2.52T + 19T^{2} \)
23 \( 1 - 0.631T + 23T^{2} \)
29 \( 1 + 7.48T + 29T^{2} \)
31 \( 1 - 7.79T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 - 7.20T + 41T^{2} \)
47 \( 1 + 9.32T + 47T^{2} \)
53 \( 1 + 4.94T + 53T^{2} \)
59 \( 1 - 1.57T + 59T^{2} \)
61 \( 1 + 6.99T + 61T^{2} \)
67 \( 1 + 3.51T + 67T^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 - 1.76T + 73T^{2} \)
79 \( 1 - 8.00T + 79T^{2} \)
83 \( 1 + 4.96T + 83T^{2} \)
89 \( 1 - 0.890T + 89T^{2} \)
97 \( 1 - 5.89T + 97T^{2} \)
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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

−8.281152168475093471020083447696, −7.83341849327652624731572072284, −6.90694912936124427925215338581, −6.23678423951110435532912899552, −5.05830647581938641370760654905, −4.20387860193371834726128559970, −3.93157914366835380234073510550, −2.99892425702818867072882330067, −1.99234549502903730066889180998, −1.45973367727438874898308463533, 1.45973367727438874898308463533, 1.99234549502903730066889180998, 2.99892425702818867072882330067, 3.93157914366835380234073510550, 4.20387860193371834726128559970, 5.05830647581938641370760654905, 6.23678423951110435532912899552, 6.90694912936124427925215338581, 7.83341849327652624731572072284, 8.281152168475093471020083447696

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