Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 55
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.07e8·2-s − 2.35e13·3-s − 2.44e16·4-s − 1.10e19·5-s − 2.52e21·6-s − 2.64e23·7-s − 6.50e24·8-s + 3.79e26·9-s − 1.18e27·10-s + 5.45e27·11-s + 5.76e29·12-s − 3.65e30·13-s − 2.84e31·14-s + 2.59e32·15-s + 1.83e32·16-s − 2.98e33·17-s + 4.07e34·18-s + 2.14e34·19-s + 2.70e35·20-s + 6.22e36·21-s + 5.85e35·22-s − 2.26e36·23-s + 1.52e38·24-s − 1.55e38·25-s − 3.92e38·26-s − 4.81e39·27-s + 6.48e39·28-s + ⋯
L(s)  = 1  + 0.566·2-s − 1.78·3-s − 0.679·4-s − 0.662·5-s − 1.00·6-s − 1.52·7-s − 0.950·8-s + 2.17·9-s − 0.375·10-s + 0.125·11-s + 1.21·12-s − 0.849·13-s − 0.862·14-s + 1.18·15-s + 0.141·16-s − 0.433·17-s + 1.23·18-s + 0.146·19-s + 0.450·20-s + 2.71·21-s + 0.0709·22-s − 0.0807·23-s + 1.69·24-s − 0.560·25-s − 0.481·26-s − 2.08·27-s + 1.03·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(56-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+55/2) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(55\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :55/2),\ 1)$
$L(28)$  $\approx$  $0.146403$
$L(\frac12)$  $\approx$  $0.146403$
$L(\frac{57}{2})$   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 \( 1 - 1.07e8T + 3.60e16T^{2} \)
3 \( 1 + 2.35e13T + 1.74e26T^{2} \)
5 \( 1 + 1.10e19T + 2.77e38T^{2} \)
7 \( 1 + 2.64e23T + 3.02e46T^{2} \)
11 \( 1 - 5.45e27T + 1.89e57T^{2} \)
13 \( 1 + 3.65e30T + 1.84e61T^{2} \)
17 \( 1 + 2.98e33T + 4.72e67T^{2} \)
19 \( 1 - 2.14e34T + 2.14e70T^{2} \)
23 \( 1 + 2.26e36T + 7.85e74T^{2} \)
29 \( 1 + 2.17e40T + 2.70e80T^{2} \)
31 \( 1 - 6.52e40T + 1.05e82T^{2} \)
37 \( 1 + 1.90e43T + 1.78e86T^{2} \)
41 \( 1 + 2.69e44T + 5.04e88T^{2} \)
43 \( 1 + 2.28e44T + 6.93e89T^{2} \)
47 \( 1 - 1.80e45T + 9.23e91T^{2} \)
53 \( 1 + 3.76e47T + 6.84e94T^{2} \)
59 \( 1 - 6.96e48T + 2.49e97T^{2} \)
61 \( 1 - 2.06e49T + 1.56e98T^{2} \)
67 \( 1 - 8.05e49T + 2.71e100T^{2} \)
71 \( 1 - 5.14e50T + 6.59e101T^{2} \)
73 \( 1 - 2.88e50T + 3.03e102T^{2} \)
79 \( 1 + 3.83e51T + 2.34e104T^{2} \)
83 \( 1 + 4.00e52T + 3.54e105T^{2} \)
89 \( 1 + 1.38e53T + 1.64e107T^{2} \)
97 \( 1 - 6.05e54T + 1.87e109T^{2} \)
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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

−18.96290724992049252949636848494, −17.22604379166101424893126275305, −15.72373022710451144447220620411, −12.95118231308152375064742191944, −11.87075427096932695407937365486, −9.846693394851963566904888415547, −6.69359478163537418462844897959, −5.29216002292109809460747630907, −3.81047115268815694039929272110, −0.25509747719462604680015866627, 0.25509747719462604680015866627, 3.81047115268815694039929272110, 5.29216002292109809460747630907, 6.69359478163537418462844897959, 9.846693394851963566904888415547, 11.87075427096932695407937365486, 12.95118231308152375064742191944, 15.72373022710451144447220620411, 17.22604379166101424893126275305, 18.96290724992049252949636848494

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