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 − 1.16·3-s + 4-s − 5-s − 1.16·6-s − 3.85·7-s + 8-s − 1.63·9-s − 10-s + 11-s − 1.16·12-s − 3.76·13-s − 3.85·14-s + 1.16·15-s + 16-s − 5.05·17-s − 1.63·18-s − 2.28·19-s − 20-s + 4.50·21-s + 22-s − 3.15·23-s − 1.16·24-s + 25-s − 3.76·26-s + 5.41·27-s − 3.85·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.673·3-s + 0.5·4-s − 0.447·5-s − 0.476·6-s − 1.45·7-s + 0.353·8-s − 0.546·9-s − 0.316·10-s + 0.301·11-s − 0.336·12-s − 1.04·13-s − 1.03·14-s + 0.301·15-s + 0.250·16-s − 1.22·17-s − 0.386·18-s − 0.525·19-s − 0.223·20-s + 0.982·21-s + 0.213·22-s − 0.658·23-s − 0.238·24-s + 0.200·25-s − 0.738·26-s + 1.04·27-s − 0.729·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8087171664$
$L(\frac12)$  $\approx$  $0.8087171664$
$L(\frac{3}{2})$   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 \( 1 + 1.16T + 3T^{2} \)
7 \( 1 + 3.85T + 7T^{2} \)
13 \( 1 + 3.76T + 13T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 + 2.28T + 19T^{2} \)
23 \( 1 + 3.15T + 23T^{2} \)
29 \( 1 - 8.35T + 29T^{2} \)
31 \( 1 + 3.01T + 31T^{2} \)
37 \( 1 + 4.67T + 37T^{2} \)
41 \( 1 + 9.20T + 41T^{2} \)
47 \( 1 - 6.69T + 47T^{2} \)
53 \( 1 - 4.13T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 7.94T + 61T^{2} \)
67 \( 1 - 7.87T + 67T^{2} \)
71 \( 1 + 0.268T + 71T^{2} \)
73 \( 1 + 3.06T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 9.21T + 83T^{2} \)
89 \( 1 + 4.61T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−8.390966813611691551582487356360, −7.02024296772372066633221579842, −6.86009992666465857793024361351, −6.11397653782986215230030194932, −5.39982161220513748027402385141, −4.57729339285734707995611765627, −3.85088013505766089229150550556, −2.97132858954179964377092749564, −2.24186852053164280242766350630, −0.42768836711532464038692326250, 0.42768836711532464038692326250, 2.24186852053164280242766350630, 2.97132858954179964377092749564, 3.85088013505766089229150550556, 4.57729339285734707995611765627, 5.39982161220513748027402385141, 6.11397653782986215230030194932, 6.86009992666465857793024361351, 7.02024296772372066633221579842, 8.390966813611691551582487356360

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