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Properties

Label 2-2001-2001.1655-c0-0-2
Degree $2$
Conductor $2001$
Sign $-0.985 - 0.168i$
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  + (−0.0895 + 0.794i)2-s + (0.0747 + 0.997i)3-s + (0.351 + 0.0801i)4-s + (−0.799 − 0.0299i)6-s + (−0.359 + 1.02i)8-s + (−0.988 + 0.149i)9-s + (−0.0537 + 0.356i)12-s + (−0.317 + 0.658i)13-s + (−0.459 − 0.221i)16-s + (−0.0299 − 0.799i)18-s + (0.781 + 0.623i)23-s + (−1.05 − 0.281i)24-s + (−0.222 + 0.974i)25-s + (−0.494 − 0.310i)26-s + (−0.222 − 0.974i)27-s + ⋯
L(s)  = 1  + (−0.0895 + 0.794i)2-s + (0.0747 + 0.997i)3-s + (0.351 + 0.0801i)4-s + (−0.799 − 0.0299i)6-s + (−0.359 + 1.02i)8-s + (−0.988 + 0.149i)9-s + (−0.0537 + 0.356i)12-s + (−0.317 + 0.658i)13-s + (−0.459 − 0.221i)16-s + (−0.0299 − 0.799i)18-s + (0.781 + 0.623i)23-s + (−1.05 − 0.281i)24-s + (−0.222 + 0.974i)25-s + (−0.494 − 0.310i)26-s + (−0.222 − 0.974i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.985 - 0.168i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ -0.985 - 0.168i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152489935\)
\(L(\frac12)\) \(\approx\) \(1.152489935\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0747 - 0.997i)T \)
23 \( 1 + (-0.781 - 0.623i)T \)
29 \( 1 + (0.294 + 0.955i)T \)
good2 \( 1 + (0.0895 - 0.794i)T + (-0.974 - 0.222i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
7 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (0.781 - 0.623i)T^{2} \)
13 \( 1 + (0.317 - 0.658i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.433 - 0.900i)T^{2} \)
31 \( 1 + (0.514 + 0.0579i)T + (0.974 + 0.222i)T^{2} \)
37 \( 1 + (-0.781 - 0.623i)T^{2} \)
41 \( 1 + (-0.262 + 0.262i)T - iT^{2} \)
43 \( 1 + (0.974 - 0.222i)T^{2} \)
47 \( 1 + (-0.882 + 0.308i)T + (0.781 - 0.623i)T^{2} \)
53 \( 1 + (-0.222 + 0.974i)T^{2} \)
59 \( 1 - 0.445iT - T^{2} \)
61 \( 1 + (-0.433 - 0.900i)T^{2} \)
67 \( 1 + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.268 - 0.129i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.98 + 0.223i)T + (0.974 - 0.222i)T^{2} \)
79 \( 1 + (0.781 + 0.623i)T^{2} \)
83 \( 1 + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.974 + 0.222i)T^{2} \)
97 \( 1 + (-0.433 + 0.900i)T^{2} \)
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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

−9.404770488876669870506900390888, −9.064529416586164641368245760422, −8.065596737565229473446160605424, −7.42027130825560759604788355366, −6.57611388137076363388643505256, −5.68412020816256078951651383099, −5.09994636189281338197051408252, −4.06632416495462339276453051905, −3.11378554876797683354594221801, −2.06193309976002374273494204137, 0.819071870434695818069593075108, 1.99016940123758317434931147849, 2.77499163968136323642875269299, 3.59283242077992209417360234200, 4.98835999755950484372312880056, 5.97630948050890846086783574243, 6.69932027797087945101779177890, 7.35954803422364083451536882719, 8.177861133105382535003573491971, 9.014435009912139035612769234218

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