# Properties

 Label 2-2001-1.1-c1-0-91 Degree $2$ Conductor $2001$ Sign $-1$ Analytic cond. $15.9780$ Root an. cond. $3.99725$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.44·2-s + 3-s + 0.0738·4-s + 0.734·5-s − 1.44·6-s + 3.73·7-s + 2.77·8-s + 9-s − 1.05·10-s − 2.70·11-s + 0.0738·12-s − 2.87·13-s − 5.38·14-s + 0.734·15-s − 4.14·16-s − 2.75·17-s − 1.44·18-s − 5.54·19-s + 0.0542·20-s + 3.73·21-s + 3.89·22-s + 23-s + 2.77·24-s − 4.46·25-s + 4.13·26-s + 27-s + 0.275·28-s + ⋯
 L(s)  = 1 − 1.01·2-s + 0.577·3-s + 0.0369·4-s + 0.328·5-s − 0.587·6-s + 1.41·7-s + 0.980·8-s + 0.333·9-s − 0.334·10-s − 0.814·11-s + 0.0213·12-s − 0.796·13-s − 1.43·14-s + 0.189·15-s − 1.03·16-s − 0.667·17-s − 0.339·18-s − 1.27·19-s + 0.0121·20-s + 0.815·21-s + 0.829·22-s + 0.208·23-s + 0.566·24-s − 0.892·25-s + 0.811·26-s + 0.192·27-s + 0.0521·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2001$$    =    $$3 \cdot 23 \cdot 29$$ Sign: $-1$ Analytic conductor: $$15.9780$$ Root analytic conductor: $$3.99725$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2001} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2001,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - T$$
23 $$1 - T$$
29 $$1 + T$$
good2 $$1 + 1.44T + 2T^{2}$$
5 $$1 - 0.734T + 5T^{2}$$
7 $$1 - 3.73T + 7T^{2}$$
11 $$1 + 2.70T + 11T^{2}$$
13 $$1 + 2.87T + 13T^{2}$$
17 $$1 + 2.75T + 17T^{2}$$
19 $$1 + 5.54T + 19T^{2}$$
31 $$1 + 10.2T + 31T^{2}$$
37 $$1 + 11.4T + 37T^{2}$$
41 $$1 + 1.81T + 41T^{2}$$
43 $$1 - 0.181T + 43T^{2}$$
47 $$1 + 6.02T + 47T^{2}$$
53 $$1 + 0.662T + 53T^{2}$$
59 $$1 + 0.174T + 59T^{2}$$
61 $$1 + 5.37T + 61T^{2}$$
67 $$1 - 15.9T + 67T^{2}$$
71 $$1 - 12.0T + 71T^{2}$$
73 $$1 - 0.990T + 73T^{2}$$
79 $$1 + 2.96T + 79T^{2}$$
83 $$1 - 1.08T + 83T^{2}$$
89 $$1 - 5.21T + 89T^{2}$$
97 $$1 + 3.97T + 97T^{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

−8.717118578015768752766380862944, −8.131549011821224595789246283597, −7.57710162090899735183185396189, −6.79547412556635043790966842538, −5.31282909916464794060561828894, −4.80422054537689023147445070983, −3.81429446373271666524148456148, −2.15497642808473395639846435875, −1.80691191805346908965086357817, 0, 1.80691191805346908965086357817, 2.15497642808473395639846435875, 3.81429446373271666524148456148, 4.80422054537689023147445070983, 5.31282909916464794060561828894, 6.79547412556635043790966842538, 7.57710162090899735183185396189, 8.131549011821224595789246283597, 8.717118578015768752766380862944