Properties

Degree $2$
Conductor $1997$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.76·2-s − 4.70·3-s − 0.381·4-s − 6.86·5-s + 12.9·6-s + 2.49·7-s + 23.1·8-s − 4.86·9-s + 18.9·10-s − 15.4·11-s + 1.79·12-s + 54.8·13-s − 6.89·14-s + 32.3·15-s − 60.8·16-s − 123.·17-s + 13.4·18-s − 0.288·19-s + 2.61·20-s − 11.7·21-s + 42.6·22-s + 33.4·23-s − 108.·24-s − 77.8·25-s − 151.·26-s + 149.·27-s − 0.953·28-s + ⋯
L(s)  = 1  − 0.975·2-s − 0.905·3-s − 0.0476·4-s − 0.614·5-s + 0.883·6-s + 0.134·7-s + 1.02·8-s − 0.180·9-s + 0.599·10-s − 0.423·11-s + 0.0431·12-s + 1.17·13-s − 0.131·14-s + 0.556·15-s − 0.950·16-s − 1.75·17-s + 0.175·18-s − 0.00348·19-s + 0.0292·20-s − 0.122·21-s + 0.413·22-s + 0.303·23-s − 0.925·24-s − 0.622·25-s − 1.14·26-s + 1.06·27-s − 0.00643·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1997 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1997\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{1997} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1997,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1997 \( 1 - 1.99e3T \)
good2 \( 1 + 2.76T + 8T^{2} \)
3 \( 1 + 4.70T + 27T^{2} \)
5 \( 1 + 6.86T + 125T^{2} \)
7 \( 1 - 2.49T + 343T^{2} \)
11 \( 1 + 15.4T + 1.33e3T^{2} \)
13 \( 1 - 54.8T + 2.19e3T^{2} \)
17 \( 1 + 123.T + 4.91e3T^{2} \)
19 \( 1 + 0.288T + 6.85e3T^{2} \)
23 \( 1 - 33.4T + 1.21e4T^{2} \)
29 \( 1 - 31.0T + 2.43e4T^{2} \)
31 \( 1 - 303.T + 2.97e4T^{2} \)
37 \( 1 + 339.T + 5.06e4T^{2} \)
41 \( 1 + 224.T + 6.89e4T^{2} \)
43 \( 1 + 176.T + 7.95e4T^{2} \)
47 \( 1 + 79.8T + 1.03e5T^{2} \)
53 \( 1 - 528.T + 1.48e5T^{2} \)
59 \( 1 + 67.2T + 2.05e5T^{2} \)
61 \( 1 + 545.T + 2.26e5T^{2} \)
67 \( 1 - 422.T + 3.00e5T^{2} \)
71 \( 1 - 948.T + 3.57e5T^{2} \)
73 \( 1 + 910.T + 3.89e5T^{2} \)
79 \( 1 - 906.T + 4.93e5T^{2} \)
83 \( 1 - 845.T + 5.71e5T^{2} \)
89 \( 1 - 592.T + 7.04e5T^{2} \)
97 \( 1 + 707.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.546552524726724736172892810631, −7.88672017683337396505804671405, −6.82161978726494705651326061007, −6.26275613539679805656083322372, −5.08169007859430710802939562944, −4.51952251621973848474094267301, −3.44991505024245830254899605747, −1.99449030327809782787160830622, −0.77735960387847036846863989937, 0, 0.77735960387847036846863989937, 1.99449030327809782787160830622, 3.44991505024245830254899605747, 4.51952251621973848474094267301, 5.08169007859430710802939562944, 6.26275613539679805656083322372, 6.82161978726494705651326061007, 7.88672017683337396505804671405, 8.546552524726724736172892810631

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