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 + 1.56·3-s + 4-s − 5-s − 1.56·6-s − 4.55·7-s − 8-s − 0.544·9-s + 10-s − 11-s + 1.56·12-s − 4.63·13-s + 4.55·14-s − 1.56·15-s + 16-s + 2.48·17-s + 0.544·18-s + 0.136·19-s − 20-s − 7.13·21-s + 22-s − 6.63·23-s − 1.56·24-s + 25-s + 4.63·26-s − 5.55·27-s − 4.55·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.904·3-s + 0.5·4-s − 0.447·5-s − 0.639·6-s − 1.72·7-s − 0.353·8-s − 0.181·9-s + 0.316·10-s − 0.301·11-s + 0.452·12-s − 1.28·13-s + 1.21·14-s − 0.404·15-s + 0.250·16-s + 0.602·17-s + 0.128·18-s + 0.0313·19-s − 0.223·20-s − 1.55·21-s + 0.213·22-s − 1.38·23-s − 0.319·24-s + 0.200·25-s + 0.908·26-s − 1.06·27-s − 0.860·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6709373509$
$L(\frac12)$  $\approx$  $0.6709373509$
$L(\frac{3}{2})$   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 \( 1 - 1.56T + 3T^{2} \)
7 \( 1 + 4.55T + 7T^{2} \)
13 \( 1 + 4.63T + 13T^{2} \)
17 \( 1 - 2.48T + 17T^{2} \)
19 \( 1 - 0.136T + 19T^{2} \)
23 \( 1 + 6.63T + 23T^{2} \)
29 \( 1 - 7.05T + 29T^{2} \)
31 \( 1 - 0.167T + 31T^{2} \)
37 \( 1 + 8.45T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
47 \( 1 - 2.16T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 0.387T + 59T^{2} \)
61 \( 1 + 6.33T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + 3.26T + 71T^{2} \)
73 \( 1 + 7.40T + 73T^{2} \)
79 \( 1 + 2.22T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 2.62T + 89T^{2} \)
97 \( 1 + 6.95T + 97T^{2} \)
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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

−8.254017479563812785825419275181, −7.72015012258075278625328627024, −7.08675487529436005279204837106, −6.29794928118006246550657454848, −5.57948519394555243671071842948, −4.35067834166714827551969709158, −3.36648444966361004784544354713, −2.90267638650518219579133040331, −2.14709898995861867837113293022, −0.44466346430493314433545081877, 0.44466346430493314433545081877, 2.14709898995861867837113293022, 2.90267638650518219579133040331, 3.36648444966361004784544354713, 4.35067834166714827551969709158, 5.57948519394555243671071842948, 6.29794928118006246550657454848, 7.08675487529436005279204837106, 7.72015012258075278625328627024, 8.254017479563812785825419275181

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