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  + 2.40e6·2-s − 6.65e9·3-s + 3.56e12·4-s − 3.38e14·5-s − 1.59e16·6-s − 9.19e16·7-s + 3.27e18·8-s + 7.87e18·9-s − 8.11e20·10-s + 2.15e21·11-s − 2.37e22·12-s − 8.71e22·13-s − 2.20e23·14-s + 2.25e24·15-s + 3.48e22·16-s + 5.13e24·17-s + 1.89e25·18-s + 4.11e25·19-s − 1.20e27·20-s + 6.12e26·21-s + 5.18e27·22-s + 2.60e27·23-s − 2.18e28·24-s + 6.88e28·25-s − 2.09e29·26-s + 1.90e29·27-s − 3.27e29·28-s + ⋯
L(s)  = 1  + 1.61·2-s − 1.10·3-s + 1.62·4-s − 1.58·5-s − 1.78·6-s − 0.435·7-s + 1.00·8-s + 0.215·9-s − 2.56·10-s + 0.967·11-s − 1.78·12-s − 1.27·13-s − 0.705·14-s + 1.74·15-s + 0.00721·16-s + 0.306·17-s + 0.349·18-s + 0.251·19-s − 2.57·20-s + 0.480·21-s + 1.56·22-s + 0.316·23-s − 1.10·24-s + 1.51·25-s − 2.05·26-s + 0.864·27-s − 0.706·28-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 \( 1 - 2.40e6T + 2.19e12T^{2} \)
3 \( 1 + 6.65e9T + 3.64e19T^{2} \)
5 \( 1 + 3.38e14T + 4.54e28T^{2} \)
7 \( 1 + 9.19e16T + 4.45e34T^{2} \)
11 \( 1 - 2.15e21T + 4.97e42T^{2} \)
13 \( 1 + 8.71e22T + 4.69e45T^{2} \)
17 \( 1 - 5.13e24T + 2.80e50T^{2} \)
19 \( 1 - 4.11e25T + 2.68e52T^{2} \)
23 \( 1 - 2.60e27T + 6.77e55T^{2} \)
29 \( 1 + 4.19e29T + 9.08e59T^{2} \)
31 \( 1 + 2.68e30T + 1.39e61T^{2} \)
37 \( 1 + 1.06e32T + 1.97e64T^{2} \)
41 \( 1 + 1.44e33T + 1.33e66T^{2} \)
43 \( 1 + 7.60e31T + 9.38e66T^{2} \)
47 \( 1 + 3.36e34T + 3.59e68T^{2} \)
53 \( 1 - 3.75e35T + 4.95e70T^{2} \)
59 \( 1 + 6.72e34T + 4.02e72T^{2} \)
61 \( 1 + 2.74e35T + 1.57e73T^{2} \)
67 \( 1 - 2.47e37T + 7.39e74T^{2} \)
71 \( 1 + 1.25e38T + 7.97e75T^{2} \)
73 \( 1 - 2.44e38T + 2.49e76T^{2} \)
79 \( 1 - 6.92e38T + 6.34e77T^{2} \)
83 \( 1 - 1.80e39T + 4.81e78T^{2} \)
89 \( 1 + 7.15e39T + 8.41e79T^{2} \)
97 \( 1 - 7.50e40T + 2.86e81T^{2} \)
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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

−22.23393646813687071807869396664, −19.75391418162021441356110899221, −16.49379218817783972206351253264, −14.86679917546178749798549819934, −12.31317004652992076923206127811, −11.51375731605963753238051158912, −6.86604869845762650166994838645, −5.00816212639601221312502909711, −3.53110805246023592341654273008, 0, 3.53110805246023592341654273008, 5.00816212639601221312502909711, 6.86604869845762650166994838645, 11.51375731605963753238051158912, 12.31317004652992076923206127811, 14.86679917546178749798549819934, 16.49379218817783972206351253264, 19.75391418162021441356110899221, 22.23393646813687071807869396664

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