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  − 0.327·2-s − 2.36·3-s − 1.89·4-s + 5-s + 0.774·6-s + 2.46·7-s + 1.27·8-s + 2.59·9-s − 0.327·10-s + 11-s + 4.47·12-s − 0.641·13-s − 0.806·14-s − 2.36·15-s + 3.36·16-s − 6.07·17-s − 0.848·18-s + 8.35·19-s − 1.89·20-s − 5.82·21-s − 0.327·22-s + 2.42·23-s − 3.01·24-s + 25-s + 0.209·26-s + 0.966·27-s − 4.66·28-s + ⋯
L(s)  = 1  − 0.231·2-s − 1.36·3-s − 0.946·4-s + 0.447·5-s + 0.316·6-s + 0.930·7-s + 0.450·8-s + 0.863·9-s − 0.103·10-s + 0.301·11-s + 1.29·12-s − 0.177·13-s − 0.215·14-s − 0.610·15-s + 0.842·16-s − 1.47·17-s − 0.199·18-s + 1.91·19-s − 0.423·20-s − 1.27·21-s − 0.0698·22-s + 0.505·23-s − 0.615·24-s + 0.200·25-s + 0.0411·26-s + 0.186·27-s − 0.880·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9797564728$
$L(\frac12)$  $\approx$  $0.9797564728$
$L(\frac{3}{2})$   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 \( 1 + 0.327T + 2T^{2} \)
3 \( 1 + 2.36T + 3T^{2} \)
7 \( 1 - 2.46T + 7T^{2} \)
13 \( 1 + 0.641T + 13T^{2} \)
17 \( 1 + 6.07T + 17T^{2} \)
19 \( 1 - 8.35T + 19T^{2} \)
23 \( 1 - 2.42T + 23T^{2} \)
29 \( 1 - 5.36T + 29T^{2} \)
31 \( 1 - 7.23T + 31T^{2} \)
37 \( 1 + 5.09T + 37T^{2} \)
41 \( 1 + 0.993T + 41T^{2} \)
43 \( 1 - 5.57T + 43T^{2} \)
47 \( 1 - 3.08T + 47T^{2} \)
53 \( 1 - 3.89T + 53T^{2} \)
59 \( 1 + 0.200T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 0.323T + 71T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 - 8.38T + 89T^{2} \)
97 \( 1 + 1.74T + 97T^{2} \)
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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

−8.587920577608856006287125409547, −7.68829163844504233029064650651, −6.90977154711079396997592666344, −6.10379122487653383427654972973, −5.32772573664235098054098971909, −4.81863401600673260943860855245, −4.32251689170461333335423355086, −2.93937369047133509769447266100, −1.49379685903680720841756092371, −0.70063625465071055124254235533, 0.70063625465071055124254235533, 1.49379685903680720841756092371, 2.93937369047133509769447266100, 4.32251689170461333335423355086, 4.81863401600673260943860855245, 5.32772573664235098054098971909, 6.10379122487653383427654972973, 6.90977154711079396997592666344, 7.68829163844504233029064650651, 8.587920577608856006287125409547

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