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 + 2.15·3-s + 4-s − 5-s + 2.15·6-s + 5.20·7-s + 8-s + 1.64·9-s − 10-s + 11-s + 2.15·12-s − 3.69·13-s + 5.20·14-s − 2.15·15-s + 16-s + 2.28·17-s + 1.64·18-s + 3.91·19-s − 20-s + 11.2·21-s + 22-s − 7.46·23-s + 2.15·24-s + 25-s − 3.69·26-s − 2.92·27-s + 5.20·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.24·3-s + 0.5·4-s − 0.447·5-s + 0.879·6-s + 1.96·7-s + 0.353·8-s + 0.547·9-s − 0.316·10-s + 0.301·11-s + 0.621·12-s − 1.02·13-s + 1.38·14-s − 0.556·15-s + 0.250·16-s + 0.555·17-s + 0.386·18-s + 0.897·19-s − 0.223·20-s + 2.44·21-s + 0.213·22-s − 1.55·23-s + 0.439·24-s + 0.200·25-s − 0.724·26-s − 0.563·27-s + 0.982·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.729274703$
$L(\frac12)$  $\approx$  $5.729274703$
$L(\frac{3}{2})$   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 \( 1 - 2.15T + 3T^{2} \)
7 \( 1 - 5.20T + 7T^{2} \)
13 \( 1 + 3.69T + 13T^{2} \)
17 \( 1 - 2.28T + 17T^{2} \)
19 \( 1 - 3.91T + 19T^{2} \)
23 \( 1 + 7.46T + 23T^{2} \)
29 \( 1 - 8.04T + 29T^{2} \)
31 \( 1 + 2.87T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 - 3.12T + 41T^{2} \)
47 \( 1 + 9.31T + 47T^{2} \)
53 \( 1 + 0.670T + 53T^{2} \)
59 \( 1 - 6.55T + 59T^{2} \)
61 \( 1 - 7.70T + 61T^{2} \)
67 \( 1 + 4.29T + 67T^{2} \)
71 \( 1 + 8.54T + 71T^{2} \)
73 \( 1 + 9.80T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 6.18T + 83T^{2} \)
89 \( 1 + 5.34T + 89T^{2} \)
97 \( 1 - 5.40T + 97T^{2} \)
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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

−8.003824207051334077188994020734, −7.81841627456593421124539964541, −7.16488784576620374672980352314, −5.94526286761694397523269975617, −5.11857666127084774466248714741, −4.46452603020322755326199427041, −3.86264456965541670131384762016, −2.84260439365552681805021233944, −2.18773237801882279681514706146, −1.25686867615493973001189647160, 1.25686867615493973001189647160, 2.18773237801882279681514706146, 2.84260439365552681805021233944, 3.86264456965541670131384762016, 4.46452603020322755326199427041, 5.11857666127084774466248714741, 5.94526286761694397523269975617, 7.16488784576620374672980352314, 7.81841627456593421124539964541, 8.003824207051334077188994020734

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