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  + 0.132·2-s + 1.97·3-s − 1.98·4-s + 5-s + 0.261·6-s + 2.54·7-s − 0.528·8-s + 0.882·9-s + 0.132·10-s + 11-s − 3.90·12-s + 3.82·13-s + 0.337·14-s + 1.97·15-s + 3.89·16-s + 3.23·17-s + 0.117·18-s + 3.88·19-s − 1.98·20-s + 5.01·21-s + 0.132·22-s − 0.671·23-s − 1.04·24-s + 25-s + 0.508·26-s − 4.17·27-s − 5.04·28-s + ⋯
L(s)  = 1  + 0.0938·2-s + 1.13·3-s − 0.991·4-s + 0.447·5-s + 0.106·6-s + 0.961·7-s − 0.186·8-s + 0.294·9-s + 0.0419·10-s + 0.301·11-s − 1.12·12-s + 1.06·13-s + 0.0902·14-s + 0.508·15-s + 0.973·16-s + 0.784·17-s + 0.0275·18-s + 0.890·19-s − 0.443·20-s + 1.09·21-s + 0.0282·22-s − 0.139·23-s − 0.212·24-s + 0.200·25-s + 0.0996·26-s − 0.803·27-s − 0.953·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.239256619$
$L(\frac12)$  $\approx$  $3.239256619$
$L(\frac{3}{2})$   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 \( 1 - 0.132T + 2T^{2} \)
3 \( 1 - 1.97T + 3T^{2} \)
7 \( 1 - 2.54T + 7T^{2} \)
13 \( 1 - 3.82T + 13T^{2} \)
17 \( 1 - 3.23T + 17T^{2} \)
19 \( 1 - 3.88T + 19T^{2} \)
23 \( 1 + 0.671T + 23T^{2} \)
29 \( 1 + 4.15T + 29T^{2} \)
31 \( 1 + 1.65T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 + 5.91T + 41T^{2} \)
43 \( 1 + 8.13T + 43T^{2} \)
47 \( 1 + 2.25T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 - 4.81T + 59T^{2} \)
61 \( 1 - 4.97T + 61T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 - 9.53T + 71T^{2} \)
79 \( 1 + 8.85T + 79T^{2} \)
83 \( 1 + 1.15T + 83T^{2} \)
89 \( 1 + 1.34T + 89T^{2} \)
97 \( 1 - 6.93T + 97T^{2} \)
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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

−8.465454456843817728005277449568, −8.005457915175666220689395559891, −7.28063334367972877207220086833, −5.99443184501317715261590313771, −5.44608133896820973723041985253, −4.58775303948778525054670698528, −3.69637125046021428182048942969, −3.17765898493622498752980081576, −1.93619340637098011812442273587, −1.06211601039443412770602836533, 1.06211601039443412770602836533, 1.93619340637098011812442273587, 3.17765898493622498752980081576, 3.69637125046021428182048942969, 4.58775303948778525054670698528, 5.44608133896820973723041985253, 5.99443184501317715261590313771, 7.28063334367972877207220086833, 8.005457915175666220689395559891, 8.465454456843817728005277449568

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