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  + 2.44·2-s − 2.95·3-s + 3.95·4-s + 5-s − 7.21·6-s − 2.50·7-s + 4.78·8-s + 5.72·9-s + 2.44·10-s + 11-s − 11.6·12-s + 4.03·13-s − 6.11·14-s − 2.95·15-s + 3.75·16-s − 4.47·17-s + 13.9·18-s + 2.04·19-s + 3.95·20-s + 7.40·21-s + 2.44·22-s − 5.54·23-s − 14.1·24-s + 25-s + 9.84·26-s − 8.05·27-s − 9.91·28-s + ⋯
L(s)  = 1  + 1.72·2-s − 1.70·3-s + 1.97·4-s + 0.447·5-s − 2.94·6-s − 0.946·7-s + 1.69·8-s + 1.90·9-s + 0.771·10-s + 0.301·11-s − 3.37·12-s + 1.11·13-s − 1.63·14-s − 0.762·15-s + 0.938·16-s − 1.08·17-s + 3.29·18-s + 0.468·19-s + 0.885·20-s + 1.61·21-s + 0.520·22-s − 1.15·23-s − 2.88·24-s + 0.200·25-s + 1.93·26-s − 1.55·27-s − 1.87·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.140513407$
$L(\frac12)$  $\approx$  $3.140513407$
$L(\frac{3}{2})$   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 \( 1 - 2.44T + 2T^{2} \)
3 \( 1 + 2.95T + 3T^{2} \)
7 \( 1 + 2.50T + 7T^{2} \)
13 \( 1 - 4.03T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 2.04T + 19T^{2} \)
23 \( 1 + 5.54T + 23T^{2} \)
29 \( 1 - 6.07T + 29T^{2} \)
31 \( 1 - 8.54T + 31T^{2} \)
37 \( 1 - 4.41T + 37T^{2} \)
41 \( 1 - 4.39T + 41T^{2} \)
43 \( 1 + 6.31T + 43T^{2} \)
47 \( 1 - 5.44T + 47T^{2} \)
53 \( 1 - 0.170T + 53T^{2} \)
59 \( 1 + 0.182T + 59T^{2} \)
61 \( 1 - 8.80T + 61T^{2} \)
67 \( 1 - 3.65T + 67T^{2} \)
71 \( 1 - 2.71T + 71T^{2} \)
79 \( 1 - 4.63T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 5.44T + 89T^{2} \)
97 \( 1 - 0.364T + 97T^{2} \)
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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

−8.266743853902802629107733024082, −6.94367924071827056257345417429, −6.47582037014665066396357820445, −6.15138609351027696154103336629, −5.58956308205001602321512306277, −4.69279931744594129446608606537, −4.18605832583150291246569648458, −3.28255280156726368423821722248, −2.18287878016278950923850506794, −0.858706378796804429980702497902, 0.858706378796804429980702497902, 2.18287878016278950923850506794, 3.28255280156726368423821722248, 4.18605832583150291246569648458, 4.69279931744594129446608606537, 5.58956308205001602321512306277, 6.15138609351027696154103336629, 6.47582037014665066396357820445, 6.94367924071827056257345417429, 8.266743853902802629107733024082

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