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  + 2.15·2-s + 2.90·3-s + 2.66·4-s + 5-s + 6.27·6-s + 4.32·7-s + 1.43·8-s + 5.44·9-s + 2.15·10-s + 11-s + 7.73·12-s − 6.34·13-s + 9.34·14-s + 2.90·15-s − 2.23·16-s − 1.54·17-s + 11.7·18-s + 0.595·19-s + 2.66·20-s + 12.5·21-s + 2.15·22-s − 2.60·23-s + 4.16·24-s + 25-s − 13.6·26-s + 7.11·27-s + 11.5·28-s + ⋯
L(s)  = 1  + 1.52·2-s + 1.67·3-s + 1.33·4-s + 0.447·5-s + 2.56·6-s + 1.63·7-s + 0.506·8-s + 1.81·9-s + 0.682·10-s + 0.301·11-s + 2.23·12-s − 1.75·13-s + 2.49·14-s + 0.750·15-s − 0.558·16-s − 0.374·17-s + 2.77·18-s + 0.136·19-s + 0.595·20-s + 2.74·21-s + 0.460·22-s − 0.543·23-s + 0.849·24-s + 0.200·25-s − 2.68·26-s + 1.36·27-s + 2.17·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $9.757123647$
$L(\frac12)$  $\approx$  $9.757123647$
$L(\frac{3}{2})$   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 \( 1 - 2.15T + 2T^{2} \)
3 \( 1 - 2.90T + 3T^{2} \)
7 \( 1 - 4.32T + 7T^{2} \)
13 \( 1 + 6.34T + 13T^{2} \)
17 \( 1 + 1.54T + 17T^{2} \)
19 \( 1 - 0.595T + 19T^{2} \)
23 \( 1 + 2.60T + 23T^{2} \)
29 \( 1 - 1.72T + 29T^{2} \)
31 \( 1 + 9.13T + 31T^{2} \)
37 \( 1 - 5.42T + 37T^{2} \)
41 \( 1 - 2.42T + 41T^{2} \)
43 \( 1 - 12.9T + 43T^{2} \)
47 \( 1 + 8.37T + 47T^{2} \)
53 \( 1 + 0.00632T + 53T^{2} \)
59 \( 1 - 8.25T + 59T^{2} \)
61 \( 1 + 9.78T + 61T^{2} \)
67 \( 1 - 0.458T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 6.52T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 0.642T + 97T^{2} \)
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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

−8.364234595080415525350915330123, −7.47267113813054948307186575388, −7.26694492456499512727604633653, −6.00186267001258595275008596677, −5.15109662751668627811996368210, −4.50561143665524738208702312951, −4.02090306314660214370865199794, −2.91355905433230581203359603772, −2.29934976734601404388715840514, −1.71786646667419100266473013593, 1.71786646667419100266473013593, 2.29934976734601404388715840514, 2.91355905433230581203359603772, 4.02090306314660214370865199794, 4.50561143665524738208702312951, 5.15109662751668627811996368210, 6.00186267001258595275008596677, 7.26694492456499512727604633653, 7.47267113813054948307186575388, 8.364234595080415525350915330123

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