Properties

Degree 2
Conductor 2
Sign $1$
Motivic weight 41
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.04e6·2-s − 1.71e9·3-s + 1.09e12·4-s + 3.51e14·5-s − 1.80e15·6-s − 7.89e16·7-s + 1.15e18·8-s − 3.35e19·9-s + 3.68e20·10-s + 1.20e21·11-s − 1.88e21·12-s + 9.23e22·13-s − 8.27e22·14-s − 6.04e23·15-s + 1.20e24·16-s − 4.95e24·17-s − 3.51e25·18-s + 2.41e26·19-s + 3.86e26·20-s + 1.35e26·21-s + 1.26e27·22-s + 8.23e27·23-s − 1.98e27·24-s + 7.83e28·25-s + 9.68e28·26-s + 1.20e29·27-s − 8.67e28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.284·3-s + 0.5·4-s + 1.65·5-s − 0.201·6-s − 0.373·7-s + 0.353·8-s − 0.919·9-s + 1.16·10-s + 0.540·11-s − 0.142·12-s + 1.34·13-s − 0.264·14-s − 0.469·15-s + 0.250·16-s − 0.295·17-s − 0.649·18-s + 1.47·19-s + 0.825·20-s + 0.106·21-s + 0.382·22-s + 1.00·23-s − 0.100·24-s + 1.72·25-s + 0.953·26-s + 0.545·27-s − 0.186·28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(41\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2,\ (\ :41/2),\ 1)$
$L(21)$  $\approx$  $3.57703$
$L(\frac12)$  $\approx$  $3.57703$
$L(\frac{43}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - 1.04e6T \)
good3 \( 1 + 1.71e9T + 3.64e19T^{2} \)
5 \( 1 - 3.51e14T + 4.54e28T^{2} \)
7 \( 1 + 7.89e16T + 4.45e34T^{2} \)
11 \( 1 - 1.20e21T + 4.97e42T^{2} \)
13 \( 1 - 9.23e22T + 4.69e45T^{2} \)
17 \( 1 + 4.95e24T + 2.80e50T^{2} \)
19 \( 1 - 2.41e26T + 2.68e52T^{2} \)
23 \( 1 - 8.23e27T + 6.77e55T^{2} \)
29 \( 1 - 2.60e29T + 9.08e59T^{2} \)
31 \( 1 + 6.40e30T + 1.39e61T^{2} \)
37 \( 1 - 7.38e31T + 1.97e64T^{2} \)
41 \( 1 - 2.15e33T + 1.33e66T^{2} \)
43 \( 1 + 5.67e33T + 9.38e66T^{2} \)
47 \( 1 + 6.45e33T + 3.59e68T^{2} \)
53 \( 1 + 9.99e34T + 4.95e70T^{2} \)
59 \( 1 + 3.33e36T + 4.02e72T^{2} \)
61 \( 1 - 2.08e36T + 1.57e73T^{2} \)
67 \( 1 - 2.21e36T + 7.39e74T^{2} \)
71 \( 1 - 5.59e37T + 7.97e75T^{2} \)
73 \( 1 + 2.02e38T + 2.49e76T^{2} \)
79 \( 1 + 2.46e38T + 6.34e77T^{2} \)
83 \( 1 + 3.59e37T + 4.81e78T^{2} \)
89 \( 1 - 5.24e38T + 8.41e79T^{2} \)
97 \( 1 + 2.35e40T + 2.86e81T^{2} \)
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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

−17.96160461071481857497799814239, −16.50444697546084319802645030174, −14.26588627843624105406328000972, −13.16628907179816938592848982938, −11.12914078299579263768529450063, −9.266134554387374088129282103860, −6.39130410049944954266522517283, −5.42902154586998137447859334147, −3.08688946578043229440874758303, −1.38155187090491114534548989819, 1.38155187090491114534548989819, 3.08688946578043229440874758303, 5.42902154586998137447859334147, 6.39130410049944954266522517283, 9.266134554387374088129282103860, 11.12914078299579263768529450063, 13.16628907179816938592848982938, 14.26588627843624105406328000972, 16.50444697546084319802645030174, 17.96160461071481857497799814239

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