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  − 2.58·2-s − 0.179·3-s + 4.68·4-s + 5-s + 0.463·6-s + 2.09·7-s − 6.93·8-s − 2.96·9-s − 2.58·10-s + 11-s − 0.839·12-s − 4.99·13-s − 5.40·14-s − 0.179·15-s + 8.56·16-s + 6.36·17-s + 7.67·18-s − 4.13·19-s + 4.68·20-s − 0.375·21-s − 2.58·22-s − 6.06·23-s + 1.24·24-s + 25-s + 12.9·26-s + 1.06·27-s + 9.79·28-s + ⋯
L(s)  = 1  − 1.82·2-s − 0.103·3-s + 2.34·4-s + 0.447·5-s + 0.189·6-s + 0.790·7-s − 2.45·8-s − 0.989·9-s − 0.817·10-s + 0.301·11-s − 0.242·12-s − 1.38·13-s − 1.44·14-s − 0.0462·15-s + 2.14·16-s + 1.54·17-s + 1.80·18-s − 0.948·19-s + 1.04·20-s − 0.0818·21-s − 0.551·22-s − 1.26·23-s + 0.253·24-s + 0.200·25-s + 2.53·26-s + 0.205·27-s + 1.85·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6324508156$
$L(\frac12)$  $\approx$  $0.6324508156$
$L(\frac{3}{2})$   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 \( 1 + 2.58T + 2T^{2} \)
3 \( 1 + 0.179T + 3T^{2} \)
7 \( 1 - 2.09T + 7T^{2} \)
13 \( 1 + 4.99T + 13T^{2} \)
17 \( 1 - 6.36T + 17T^{2} \)
19 \( 1 + 4.13T + 19T^{2} \)
23 \( 1 + 6.06T + 23T^{2} \)
29 \( 1 + 8.53T + 29T^{2} \)
31 \( 1 - 0.350T + 31T^{2} \)
37 \( 1 - 0.503T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 + 3.35T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 6.15T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 - 0.730T + 67T^{2} \)
71 \( 1 - 9.27T + 71T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 2.88T + 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 - 2.56T + 97T^{2} \)
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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

−8.323881611466442444327372081876, −7.964363942175293255580013962743, −7.35654171857583720870563743871, −6.40710354538383919695888812279, −5.75079952294459293439392105804, −4.94416635332589588739216976041, −3.51510236649633131890043672347, −2.33433031685816471698800204097, −1.87689498795461994067319160312, −0.58408655663602234726426457160, 0.58408655663602234726426457160, 1.87689498795461994067319160312, 2.33433031685816471698800204097, 3.51510236649633131890043672347, 4.94416635332589588739216976041, 5.75079952294459293439392105804, 6.40710354538383919695888812279, 7.35654171857583720870563743871, 7.964363942175293255580013962743, 8.323881611466442444327372081876

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