Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 89
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.24e13·2-s − 6.44e20·3-s − 1.14e26·4-s − 1.81e31·5-s + 1.44e34·6-s − 2.76e37·7-s + 1.64e40·8-s − 2.49e42·9-s + 4.07e44·10-s + 2.89e46·11-s + 7.37e46·12-s − 2.93e49·13-s + 6.20e50·14-s + 1.16e52·15-s − 2.99e53·16-s + 3.53e54·17-s + 5.60e55·18-s − 2.73e56·19-s + 2.07e57·20-s + 1.78e58·21-s − 6.49e59·22-s + 5.99e60·23-s − 1.06e61·24-s + 1.66e62·25-s + 6.58e62·26-s + 3.48e63·27-s + 3.15e63·28-s + ⋯
L(s)  = 1  − 0.902·2-s − 0.378·3-s − 0.184·4-s − 1.42·5-s + 0.341·6-s − 0.682·7-s + 1.06·8-s − 0.857·9-s + 1.28·10-s + 1.31·11-s + 0.0698·12-s − 0.787·13-s + 0.616·14-s + 0.538·15-s − 0.781·16-s + 0.621·17-s + 0.773·18-s − 0.340·19-s + 0.263·20-s + 0.258·21-s − 1.18·22-s + 1.51·23-s − 0.404·24-s + 1.03·25-s + 0.711·26-s + 0.702·27-s + 0.126·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(90-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+89/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(89\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :89/2),\ -1)\)
\(L(45)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{91}{2})\)   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 \( 1 + 2.24e13T + 6.18e26T^{2} \)
3 \( 1 + 6.44e20T + 2.90e42T^{2} \)
5 \( 1 + 1.81e31T + 1.61e62T^{2} \)
7 \( 1 + 2.76e37T + 1.63e75T^{2} \)
11 \( 1 - 2.89e46T + 4.83e92T^{2} \)
13 \( 1 + 2.93e49T + 1.38e99T^{2} \)
17 \( 1 - 3.53e54T + 3.23e109T^{2} \)
19 \( 1 + 2.73e56T + 6.44e113T^{2} \)
23 \( 1 - 5.99e60T + 1.56e121T^{2} \)
29 \( 1 - 2.05e65T + 1.42e130T^{2} \)
31 \( 1 + 1.59e66T + 5.38e132T^{2} \)
37 \( 1 - 1.13e69T + 3.71e139T^{2} \)
41 \( 1 - 8.14e70T + 3.44e143T^{2} \)
43 \( 1 - 3.18e72T + 2.39e145T^{2} \)
47 \( 1 + 3.19e74T + 6.55e148T^{2} \)
53 \( 1 + 1.04e77T + 2.88e153T^{2} \)
59 \( 1 - 8.65e78T + 4.03e157T^{2} \)
61 \( 1 + 3.87e79T + 7.84e158T^{2} \)
67 \( 1 - 1.41e81T + 3.31e162T^{2} \)
71 \( 1 - 1.52e82T + 5.78e164T^{2} \)
73 \( 1 + 5.28e82T + 6.85e165T^{2} \)
79 \( 1 - 4.95e84T + 7.74e168T^{2} \)
83 \( 1 - 5.30e84T + 6.27e170T^{2} \)
89 \( 1 + 6.27e86T + 3.13e173T^{2} \)
97 \( 1 - 2.15e88T + 6.64e176T^{2} \)
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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

−14.53074258366583036640821778686, −12.34084534606860609110999755448, −11.09969066201492674814066400277, −9.394939964391648639794585083696, −8.182079039762662653481847604366, −6.79563336319194099738029645743, −4.68605182707309679630770282207, −3.28763863480721116545096077530, −0.917234848080778442013681445998, 0, 0.917234848080778442013681445998, 3.28763863480721116545096077530, 4.68605182707309679630770282207, 6.79563336319194099738029645743, 8.182079039762662653481847604366, 9.394939964391648639794585083696, 11.09969066201492674814066400277, 12.34084534606860609110999755448, 14.53074258366583036640821778686

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