Properties

Degree 2
Conductor $ 5 \cdot 11 \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.53·2-s + 2.11·3-s + 4.40·4-s + 5-s − 5.36·6-s + 2.19·7-s − 6.10·8-s + 1.49·9-s − 2.53·10-s + 11-s + 9.34·12-s + 4.02·13-s − 5.56·14-s + 2.11·15-s + 6.62·16-s + 6.58·17-s − 3.77·18-s − 4.95·19-s + 4.40·20-s + 4.65·21-s − 2.53·22-s − 2.67·23-s − 12.9·24-s + 25-s − 10.1·26-s − 3.19·27-s + 9.69·28-s + ⋯
L(s)  = 1  − 1.79·2-s + 1.22·3-s + 2.20·4-s + 0.447·5-s − 2.19·6-s + 0.830·7-s − 2.15·8-s + 0.497·9-s − 0.800·10-s + 0.301·11-s + 2.69·12-s + 1.11·13-s − 1.48·14-s + 0.547·15-s + 1.65·16-s + 1.59·17-s − 0.889·18-s − 1.13·19-s + 0.986·20-s + 1.01·21-s − 0.539·22-s − 0.557·23-s − 2.63·24-s + 0.200·25-s − 1.99·26-s − 0.615·27-s + 1.83·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4015\)    =    \(5 \cdot 11 \cdot 73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4015} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.804282320$
$L(\frac12)$  $\approx$  $1.804282320$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11,\;73\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;11,\;73\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - T \)
11 \( 1 - T \)
73 \( 1 - T \)
good2 \( 1 + 2.53T + 2T^{2} \)
3 \( 1 - 2.11T + 3T^{2} \)
7 \( 1 - 2.19T + 7T^{2} \)
13 \( 1 - 4.02T + 13T^{2} \)
17 \( 1 - 6.58T + 17T^{2} \)
19 \( 1 + 4.95T + 19T^{2} \)
23 \( 1 + 2.67T + 23T^{2} \)
29 \( 1 - 9.82T + 29T^{2} \)
31 \( 1 - 6.06T + 31T^{2} \)
37 \( 1 - 0.228T + 37T^{2} \)
41 \( 1 + 1.39T + 41T^{2} \)
43 \( 1 + 4.42T + 43T^{2} \)
47 \( 1 - 6.69T + 47T^{2} \)
53 \( 1 - 9.42T + 53T^{2} \)
59 \( 1 + 7.06T + 59T^{2} \)
61 \( 1 + 3.47T + 61T^{2} \)
67 \( 1 + 16.1T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
79 \( 1 - 6.89T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 5.76T + 97T^{2} \)
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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

−8.454547421520684025286221976286, −8.096559733151265623089986127542, −7.48842215192683297723307639326, −6.46191548988733362100797663916, −5.91554832089158781469782905157, −4.54421554252694994018374613918, −3.39692889911649441315538164514, −2.58651384056622940105109989886, −1.72833368371605790806563666107, −1.03731624655869460168071479060, 1.03731624655869460168071479060, 1.72833368371605790806563666107, 2.58651384056622940105109989886, 3.39692889911649441315538164514, 4.54421554252694994018374613918, 5.91554832089158781469782905157, 6.46191548988733362100797663916, 7.48842215192683297723307639326, 8.096559733151265623089986127542, 8.454547421520684025286221976286

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