Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 73
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.28e10·2-s − 4.48e17·3-s − 9.28e21·4-s + 4.05e25·5-s − 5.74e27·6-s − 7.12e30·7-s − 2.39e32·8-s + 1.33e35·9-s + 5.19e35·10-s + 5.42e37·11-s + 4.16e39·12-s + 7.16e40·13-s − 9.12e40·14-s − 1.82e43·15-s + 8.45e43·16-s + 1.30e44·17-s + 1.71e45·18-s − 2.65e46·19-s − 3.76e47·20-s + 3.19e48·21-s + 6.94e47·22-s − 6.34e49·23-s + 1.07e50·24-s + 5.89e50·25-s + 9.17e50·26-s − 2.97e52·27-s + 6.61e52·28-s + ⋯
L(s)  = 1  + 0.131·2-s − 1.72·3-s − 0.982·4-s + 1.24·5-s − 0.227·6-s − 1.01·7-s − 0.261·8-s + 1.98·9-s + 0.164·10-s + 0.529·11-s + 1.69·12-s + 1.57·13-s − 0.133·14-s − 2.15·15-s + 0.948·16-s + 0.160·17-s + 0.260·18-s − 0.561·19-s − 1.22·20-s + 1.75·21-s + 0.0697·22-s − 1.25·23-s + 0.451·24-s + 0.556·25-s + 0.206·26-s − 1.69·27-s + 0.997·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(74-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+73/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(73\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :73/2),\ -1)\)
\(L(37)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{75}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 - 1.28e10T + 9.44e21T^{2} \)
3 \( 1 + 4.48e17T + 6.75e34T^{2} \)
5 \( 1 - 4.05e25T + 1.05e51T^{2} \)
7 \( 1 + 7.12e30T + 4.92e61T^{2} \)
11 \( 1 - 5.42e37T + 1.05e76T^{2} \)
13 \( 1 - 7.16e40T + 2.07e81T^{2} \)
17 \( 1 - 1.30e44T + 6.64e89T^{2} \)
19 \( 1 + 2.65e46T + 2.23e93T^{2} \)
23 \( 1 + 6.34e49T + 2.54e99T^{2} \)
29 \( 1 - 8.00e52T + 5.68e106T^{2} \)
31 \( 1 + 3.53e54T + 7.40e108T^{2} \)
37 \( 1 - 1.61e56T + 3.01e114T^{2} \)
41 \( 1 - 6.62e58T + 5.41e117T^{2} \)
43 \( 1 - 5.13e57T + 1.75e119T^{2} \)
47 \( 1 - 1.87e61T + 1.15e122T^{2} \)
53 \( 1 + 2.36e62T + 7.44e125T^{2} \)
59 \( 1 + 2.19e64T + 1.87e129T^{2} \)
61 \( 1 + 1.30e65T + 2.13e130T^{2} \)
67 \( 1 + 5.38e66T + 2.01e133T^{2} \)
71 \( 1 - 3.88e67T + 1.38e135T^{2} \)
73 \( 1 + 9.28e67T + 1.05e136T^{2} \)
79 \( 1 + 4.44e68T + 3.36e138T^{2} \)
83 \( 1 - 9.20e69T + 1.23e140T^{2} \)
89 \( 1 + 3.27e70T + 2.02e142T^{2} \)
97 \( 1 + 2.09e72T + 1.08e145T^{2} \)
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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

−16.32397618486113917069065523289, −13.63431323517529663649049601018, −12.50701869284297717295949755892, −10.55945348728993121237476851858, −9.336242290136873608398234029104, −6.24241685386205450248069218885, −5.73303297228531362946256654245, −4.01676214092327570276345851221, −1.28192350893123960744080206343, 0, 1.28192350893123960744080206343, 4.01676214092327570276345851221, 5.73303297228531362946256654245, 6.24241685386205450248069218885, 9.336242290136873608398234029104, 10.55945348728993121237476851858, 12.50701869284297717295949755892, 13.63431323517529663649049601018, 16.32397618486113917069065523289

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