Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 41
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.66e6·2-s − 7.98e9·3-s + 4.89e12·4-s + 3.47e13·5-s + 2.12e16·6-s + 1.66e17·7-s − 7.19e18·8-s + 2.72e19·9-s − 9.25e19·10-s − 1.73e21·11-s − 3.91e22·12-s + 9.18e22·13-s − 4.44e23·14-s − 2.77e23·15-s + 8.39e24·16-s + 2.02e25·17-s − 7.26e25·18-s − 1.01e26·19-s + 1.70e26·20-s − 1.33e27·21-s + 4.62e27·22-s − 1.07e28·23-s + 5.74e28·24-s − 4.42e28·25-s − 2.44e29·26-s + 7.35e28·27-s + 8.17e29·28-s + ⋯
L(s)  = 1  − 1.79·2-s − 1.32·3-s + 2.22·4-s + 0.162·5-s + 2.37·6-s + 0.790·7-s − 2.20·8-s + 0.747·9-s − 0.292·10-s − 0.778·11-s − 2.94·12-s + 1.33·13-s − 1.41·14-s − 0.215·15-s + 1.73·16-s + 1.21·17-s − 1.34·18-s − 0.621·19-s + 0.362·20-s − 1.04·21-s + 1.39·22-s − 1.31·23-s + 2.91·24-s − 0.973·25-s − 2.40·26-s + 0.333·27-s + 1.76·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\,\Lambda(42-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(41\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1,\ (\ :41/2),\ -1)$
$L(21)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{43}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 2.
$p$$F_p$
good2 \( 1 + 2.66e6T + 2.19e12T^{2} \)
3 \( 1 + 7.98e9T + 3.64e19T^{2} \)
5 \( 1 - 3.47e13T + 4.54e28T^{2} \)
7 \( 1 - 1.66e17T + 4.45e34T^{2} \)
11 \( 1 + 1.73e21T + 4.97e42T^{2} \)
13 \( 1 - 9.18e22T + 4.69e45T^{2} \)
17 \( 1 - 2.02e25T + 2.80e50T^{2} \)
19 \( 1 + 1.01e26T + 2.68e52T^{2} \)
23 \( 1 + 1.07e28T + 6.77e55T^{2} \)
29 \( 1 + 8.13e29T + 9.08e59T^{2} \)
31 \( 1 - 2.07e30T + 1.39e61T^{2} \)
37 \( 1 + 3.24e30T + 1.97e64T^{2} \)
41 \( 1 + 1.06e33T + 1.33e66T^{2} \)
43 \( 1 - 1.71e33T + 9.38e66T^{2} \)
47 \( 1 + 9.87e33T + 3.59e68T^{2} \)
53 \( 1 + 1.57e35T + 4.95e70T^{2} \)
59 \( 1 + 1.93e36T + 4.02e72T^{2} \)
61 \( 1 - 6.18e36T + 1.57e73T^{2} \)
67 \( 1 + 4.06e37T + 7.39e74T^{2} \)
71 \( 1 + 6.45e37T + 7.97e75T^{2} \)
73 \( 1 - 3.02e37T + 2.49e76T^{2} \)
79 \( 1 + 1.38e39T + 6.34e77T^{2} \)
83 \( 1 - 1.09e39T + 4.81e78T^{2} \)
89 \( 1 - 6.39e38T + 8.41e79T^{2} \)
97 \( 1 - 3.04e39T + 2.86e81T^{2} \)
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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

−20.92918365859342746781592699153, −18.50764456138899202751180438173, −17.48033887266875676383394906611, −16.09743473091875402325370095204, −11.55581316214425608002854778495, −10.34711477872763083081943080374, −8.050931476300633943553916436279, −5.94895438381446622073970476791, −1.49703521748310075275001255323, 0, 1.49703521748310075275001255323, 5.94895438381446622073970476791, 8.050931476300633943553916436279, 10.34711477872763083081943080374, 11.55581316214425608002854778495, 16.09743473091875402325370095204, 17.48033887266875676383394906611, 18.50764456138899202751180438173, 20.92918365859342746781592699153

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