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  + 2.50·2-s + 2.36·3-s + 4.25·4-s + 5-s + 5.92·6-s − 2.67·7-s + 5.64·8-s + 2.60·9-s + 2.50·10-s + 11-s + 10.0·12-s + 3.48·13-s − 6.68·14-s + 2.36·15-s + 5.60·16-s + 4.44·17-s + 6.52·18-s − 4.04·19-s + 4.25·20-s − 6.32·21-s + 2.50·22-s + 5.55·23-s + 13.3·24-s + 25-s + 8.71·26-s − 0.930·27-s − 11.3·28-s + ⋯
L(s)  = 1  + 1.76·2-s + 1.36·3-s + 2.12·4-s + 0.447·5-s + 2.41·6-s − 1.00·7-s + 1.99·8-s + 0.868·9-s + 0.790·10-s + 0.301·11-s + 2.90·12-s + 0.966·13-s − 1.78·14-s + 0.611·15-s + 1.40·16-s + 1.07·17-s + 1.53·18-s − 0.927·19-s + 0.951·20-s − 1.38·21-s + 0.533·22-s + 1.15·23-s + 2.72·24-s + 0.200·25-s + 1.70·26-s − 0.179·27-s − 2.14·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $9.543087910$
$L(\frac12)$  $\approx$  $9.543087910$
$L(\frac{3}{2})$   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 \( 1 - 2.50T + 2T^{2} \)
3 \( 1 - 2.36T + 3T^{2} \)
7 \( 1 + 2.67T + 7T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 - 4.44T + 17T^{2} \)
19 \( 1 + 4.04T + 19T^{2} \)
23 \( 1 - 5.55T + 23T^{2} \)
29 \( 1 + 6.50T + 29T^{2} \)
31 \( 1 + 2.94T + 31T^{2} \)
37 \( 1 - 0.596T + 37T^{2} \)
41 \( 1 - 5.28T + 41T^{2} \)
43 \( 1 - 6.60T + 43T^{2} \)
47 \( 1 - 0.172T + 47T^{2} \)
53 \( 1 + 4.95T + 53T^{2} \)
59 \( 1 + 4.38T + 59T^{2} \)
61 \( 1 - 2.23T + 61T^{2} \)
67 \( 1 - 5.08T + 67T^{2} \)
71 \( 1 + 3.91T + 71T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 8.54T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + 8.71T + 97T^{2} \)
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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

−8.408004683739896929252153176512, −7.45890290881075095212017274478, −6.83013930719230012623354908234, −6.00243830223916062161775817956, −5.58101823721906115251839085500, −4.37748368780119578474178109110, −3.67310162036144320995197999909, −3.19242056566737477143776056595, −2.51769155085796141277037979361, −1.53450754409896208109093325712, 1.53450754409896208109093325712, 2.51769155085796141277037979361, 3.19242056566737477143776056595, 3.67310162036144320995197999909, 4.37748368780119578474178109110, 5.58101823721906115251839085500, 6.00243830223916062161775817956, 6.83013930719230012623354908234, 7.45890290881075095212017274478, 8.408004683739896929252153176512

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