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 + 1.95·3-s + 4-s − 5-s − 1.95·6-s + 0.00701·7-s − 8-s + 0.816·9-s + 10-s − 11-s + 1.95·12-s + 5.21·13-s − 0.00701·14-s − 1.95·15-s + 16-s + 5.04·17-s − 0.816·18-s − 5.60·19-s − 20-s + 0.0137·21-s + 22-s + 3.21·23-s − 1.95·24-s + 25-s − 5.21·26-s − 4.26·27-s + 0.00701·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.12·3-s + 0.5·4-s − 0.447·5-s − 0.797·6-s + 0.00265·7-s − 0.353·8-s + 0.272·9-s + 0.316·10-s − 0.301·11-s + 0.563·12-s + 1.44·13-s − 0.00187·14-s − 0.504·15-s + 0.250·16-s + 1.22·17-s − 0.192·18-s − 1.28·19-s − 0.223·20-s + 0.00299·21-s + 0.213·22-s + 0.671·23-s − 0.398·24-s + 0.200·25-s − 1.02·26-s − 0.820·27-s + 0.00132·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.972137574$
$L(\frac12)$  $\approx$  $1.972137574$
$L(\frac{3}{2})$   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 \( 1 - 1.95T + 3T^{2} \)
7 \( 1 - 0.00701T + 7T^{2} \)
13 \( 1 - 5.21T + 13T^{2} \)
17 \( 1 - 5.04T + 17T^{2} \)
19 \( 1 + 5.60T + 19T^{2} \)
23 \( 1 - 3.21T + 23T^{2} \)
29 \( 1 + 2.96T + 29T^{2} \)
31 \( 1 - 0.858T + 31T^{2} \)
37 \( 1 - 6.65T + 37T^{2} \)
41 \( 1 - 1.86T + 41T^{2} \)
47 \( 1 - 8.84T + 47T^{2} \)
53 \( 1 - 0.559T + 53T^{2} \)
59 \( 1 - 2.32T + 59T^{2} \)
61 \( 1 - 4.42T + 61T^{2} \)
67 \( 1 - 1.93T + 67T^{2} \)
71 \( 1 + 1.54T + 71T^{2} \)
73 \( 1 - 7.76T + 73T^{2} \)
79 \( 1 + 3.56T + 79T^{2} \)
83 \( 1 - 0.476T + 83T^{2} \)
89 \( 1 + 7.90T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−8.313322245689669466024186717082, −7.88024018324639277430878607380, −7.15896369781778951136359589227, −6.22835722223809257908593226702, −5.55388988160662269362913336262, −4.26366024662614582773950459334, −3.54330341231503032698331424570, −2.87162513644103684371178816702, −1.93845727174227136478418400532, −0.828332573334599911834531823611, 0.828332573334599911834531823611, 1.93845727174227136478418400532, 2.87162513644103684371178816702, 3.54330341231503032698331424570, 4.26366024662614582773950459334, 5.55388988160662269362913336262, 6.22835722223809257908593226702, 7.15896369781778951136359589227, 7.88024018324639277430878607380, 8.313322245689669466024186717082

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