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Properties

Label 2-2001-2001.1931-c0-0-0
Degree $2$
Conductor $2001$
Sign $0.944 - 0.327i$
Analytic cond. $0.998629$
Root an. cond. $0.999314$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 − 0.366i)2-s + (−0.5 + 0.866i)3-s − 0.732i·4-s + (0.5 − 0.133i)6-s + (−0.633 + 0.633i)8-s + (−0.499 − 0.866i)9-s + (0.633 + 0.366i)12-s + i·13-s − 0.267·16-s + (−0.133 + 0.499i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (0.366 − 0.366i)26-s + 0.999·27-s + ⋯
L(s)  = 1  + (−0.366 − 0.366i)2-s + (−0.5 + 0.866i)3-s − 0.732i·4-s + (0.5 − 0.133i)6-s + (−0.633 + 0.633i)8-s + (−0.499 − 0.866i)9-s + (0.633 + 0.366i)12-s + i·13-s − 0.267·16-s + (−0.133 + 0.499i)18-s + i·23-s + (−0.232 − 0.866i)24-s + 25-s + (0.366 − 0.366i)26-s + 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.944 - 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7381606809\)
\(L(\frac12)\) \(\approx\) \(0.7381606809\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + iT^{2} \)
31 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.635281251366485370352921435145, −8.904363434354504577400248109405, −8.128911597441672183842904856222, −6.67418707408202703403184723180, −6.29830716367017041866921438655, −5.18410203319351804952182956584, −4.71607887958289950662908195534, −3.60951970530572468363066648295, −2.47051027480939710241607591930, −1.10676158517570386946840708730, 0.795805904923888616144521687152, 2.45778437444825477756707638621, 3.24421403258479044037726120283, 4.57906979751940263258438758817, 5.45312103064552829472439465276, 6.53535344651417006818297298832, 6.86100795317575632897686229792, 7.79447343708667465739090896143, 8.399312595516390809794310351878, 8.877697691907082827489808676285

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