Properties

Degree 2
Conductor 2
Sign $-1$
Motivic weight 81
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.09e12·2-s + 6.69e18·3-s + 1.20e24·4-s − 3.52e28·5-s − 7.35e30·6-s − 2.28e34·7-s − 1.32e36·8-s − 3.98e38·9-s + 3.87e40·10-s + 1.33e42·11-s + 8.08e42·12-s + 1.30e45·13-s + 2.51e46·14-s − 2.35e47·15-s + 1.46e48·16-s + 1.15e50·17-s + 4.38e50·18-s + 8.59e51·19-s − 4.26e52·20-s − 1.53e53·21-s − 1.46e54·22-s − 1.84e55·23-s − 8.89e54·24-s + 8.28e56·25-s − 1.43e57·26-s − 5.63e57·27-s − 2.76e58·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.317·3-s + 0.5·4-s − 1.73·5-s − 0.224·6-s − 1.35·7-s − 0.353·8-s − 0.899·9-s + 1.22·10-s + 0.887·11-s + 0.158·12-s + 1.00·13-s + 0.960·14-s − 0.550·15-s + 0.250·16-s + 1.69·17-s + 0.635·18-s + 1.39·19-s − 0.866·20-s − 0.431·21-s − 0.627·22-s − 1.30·23-s − 0.112·24-s + 2.00·25-s − 0.708·26-s − 0.603·27-s − 0.679·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(82-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+81/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(81\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2,\ (\ :81/2),\ -1)$
$L(41)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{83}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + 1.09e12T \)
good3 \( 1 - 6.69e18T + 4.43e38T^{2} \)
5 \( 1 + 3.52e28T + 4.13e56T^{2} \)
7 \( 1 + 2.28e34T + 2.83e68T^{2} \)
11 \( 1 - 1.33e42T + 2.25e84T^{2} \)
13 \( 1 - 1.30e45T + 1.69e90T^{2} \)
17 \( 1 - 1.15e50T + 4.63e99T^{2} \)
19 \( 1 - 8.59e51T + 3.79e103T^{2} \)
23 \( 1 + 1.84e55T + 1.99e110T^{2} \)
29 \( 1 - 1.25e59T + 2.84e118T^{2} \)
31 \( 1 + 3.18e60T + 6.31e120T^{2} \)
37 \( 1 - 8.97e62T + 1.05e127T^{2} \)
41 \( 1 - 2.04e65T + 4.32e130T^{2} \)
43 \( 1 + 1.63e66T + 2.04e132T^{2} \)
47 \( 1 - 2.78e66T + 2.75e135T^{2} \)
53 \( 1 + 7.25e69T + 4.63e139T^{2} \)
59 \( 1 + 7.45e70T + 2.74e143T^{2} \)
61 \( 1 - 7.37e71T + 4.08e144T^{2} \)
67 \( 1 - 1.64e73T + 8.16e147T^{2} \)
71 \( 1 - 1.42e74T + 8.95e149T^{2} \)
73 \( 1 - 7.91e74T + 8.49e150T^{2} \)
79 \( 1 + 6.52e76T + 5.10e153T^{2} \)
83 \( 1 + 4.61e77T + 2.78e155T^{2} \)
89 \( 1 - 2.48e77T + 7.95e157T^{2} \)
97 \( 1 + 1.43e79T + 8.48e160T^{2} \)
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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

−12.26653057632648947503604842573, −11.39808348000426711159930058706, −9.679392999521842746831868360653, −8.435405160896127290094455419338, −7.43518670554110222039130674780, −6.01276076126792328979619132152, −3.60622958348746428171113004427, −3.24056467226841452092943992394, −1.00739705264753796023107134327, 0, 1.00739705264753796023107134327, 3.24056467226841452092943992394, 3.60622958348746428171113004427, 6.01276076126792328979619132152, 7.43518670554110222039130674780, 8.435405160896127290094455419338, 9.679392999521842746831868360653, 11.39808348000426711159930058706, 12.26653057632648947503604842573

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