Properties

Label 2-2001-2001.275-c0-0-4
Degree $2$
Conductor $2001$
Sign $0.805 + 0.592i$
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  + (1.82 − 0.205i)2-s + (−0.930 − 0.365i)3-s + (2.30 − 0.525i)4-s + (−1.77 − 0.474i)6-s + (2.35 − 0.823i)8-s + (0.733 + 0.680i)9-s + (−2.33 − 0.351i)12-s + (0.317 + 0.658i)13-s + (1.99 − 0.959i)16-s + (1.47 + 1.08i)18-s + (0.781 − 0.623i)23-s + (−2.49 − 0.0932i)24-s + (−0.222 − 0.974i)25-s + (0.712 + 1.13i)26-s + (−0.433 − 0.900i)27-s + ⋯
L(s)  = 1  + (1.82 − 0.205i)2-s + (−0.930 − 0.365i)3-s + (2.30 − 0.525i)4-s + (−1.77 − 0.474i)6-s + (2.35 − 0.823i)8-s + (0.733 + 0.680i)9-s + (−2.33 − 0.351i)12-s + (0.317 + 0.658i)13-s + (1.99 − 0.959i)16-s + (1.47 + 1.08i)18-s + (0.781 − 0.623i)23-s + (−2.49 − 0.0932i)24-s + (−0.222 − 0.974i)25-s + (0.712 + 1.13i)26-s + (−0.433 − 0.900i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(0.998629\)
Root analytic conductor: \(0.999314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2001,\ (\ :0),\ 0.805 + 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.642262976\)
\(L(\frac12)\) \(\approx\) \(2.642262976\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.930 + 0.365i)T \)
23 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 + (0.294 - 0.955i)T \)
good2 \( 1 + (-1.82 + 0.205i)T + (0.974 - 0.222i)T^{2} \)
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 \( 1 + (0.216 + 1.91i)T + (-0.974 + 0.222i)T^{2} \)
37 \( 1 + (0.781 - 0.623i)T^{2} \)
41 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
43 \( 1 + (-0.974 - 0.222i)T^{2} \)
47 \( 1 + (0.584 - 1.66i)T + (-0.781 - 0.623i)T^{2} \)
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 \( 1 + (0.00837 - 0.0743i)T + (-0.974 - 0.222i)T^{2} \)
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

−9.550087684418830895871204217568, −8.165000453223636249354573928251, −7.23763813285487672305572681988, −6.42159348353120926849438077430, −6.13582998559637665954942798027, −5.05508702025434295849248187931, −4.59484848805067413229257657068, −3.69071427963656715122491546883, −2.53732128928200008530074292776, −1.51905882126686440199205278734, 1.67646259581008413659666997861, 3.22762176018708744032667845553, 3.70858345159086361843677538259, 4.79975521526985883995346930426, 5.33801522705563255674744714468, 5.88870203385560604385261049396, 6.86170885769018546738470132625, 7.25830496454522802851146023376, 8.499525884490213852818848947933, 9.679438643354122828066780479493

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