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Properties

Label 1-2001-2001.488-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.726 - 0.687i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (0.714 − 0.699i)2-s + (0.0203 − 0.999i)4-s + (0.488 − 0.872i)5-s + (−0.262 + 0.965i)7-s + (−0.685 − 0.728i)8-s + (−0.262 − 0.965i)10-s + (0.947 + 0.320i)11-s + (0.182 − 0.983i)13-s + (0.488 + 0.872i)14-s + (−0.999 − 0.0407i)16-s + (0.415 − 0.909i)17-s + (−0.0203 + 0.999i)19-s + (−0.862 − 0.505i)20-s + (0.900 − 0.433i)22-s + (−0.523 − 0.852i)25-s + (−0.557 − 0.830i)26-s + ⋯
L(s)  = 1  + (0.714 − 0.699i)2-s + (0.0203 − 0.999i)4-s + (0.488 − 0.872i)5-s + (−0.262 + 0.965i)7-s + (−0.685 − 0.728i)8-s + (−0.262 − 0.965i)10-s + (0.947 + 0.320i)11-s + (0.182 − 0.983i)13-s + (0.488 + 0.872i)14-s + (−0.999 − 0.0407i)16-s + (0.415 − 0.909i)17-s + (−0.0203 + 0.999i)19-s + (−0.862 − 0.505i)20-s + (0.900 − 0.433i)22-s + (−0.523 − 0.852i)25-s + (−0.557 − 0.830i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.726 - 0.687i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.726 - 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9400652986 - 2.361962624i\)
\(L(\frac12)\) \(\approx\) \(0.9400652986 - 2.361962624i\)
\(L(1)\) \(\approx\) \(1.308916088 - 1.003474504i\)
\(L(1)\) \(\approx\) \(1.308916088 - 1.003474504i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.714 - 0.699i)T \)
5 \( 1 + (0.488 - 0.872i)T \)
7 \( 1 + (-0.262 + 0.965i)T \)
11 \( 1 + (0.947 + 0.320i)T \)
13 \( 1 + (0.182 - 0.983i)T \)
17 \( 1 + (0.415 - 0.909i)T \)
19 \( 1 + (-0.0203 + 0.999i)T \)
31 \( 1 + (-0.933 - 0.359i)T \)
37 \( 1 + (0.377 - 0.925i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.933 - 0.359i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.591 - 0.806i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (0.301 + 0.953i)T \)
67 \( 1 + (-0.947 + 0.320i)T \)
71 \( 1 + (-0.101 - 0.994i)T \)
73 \( 1 + (0.794 - 0.607i)T \)
79 \( 1 + (0.999 - 0.0407i)T \)
83 \( 1 + (0.986 + 0.162i)T \)
89 \( 1 + (0.996 + 0.0815i)T \)
97 \( 1 + (-0.742 - 0.670i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.335676507970760092328463404655, −19.440810282571660423084710160458, −18.819061824612306755207769653425, −17.64914694189657255043180111303, −17.29899958483317270840211322618, −16.54501085314385032996721275316, −15.904375090512451172296482106740, −14.81204678371636804699265601360, −14.38705060575577651399868825210, −13.800838233188602757521404749116, −13.15764747115839639404199209849, −12.29912065821246819771251993901, −11.2010658448424496455847024239, −10.91824428543641356709942282396, −9.638109713861207246616441010815, −9.0599851413659190590800287422, −7.921174644326835417546236078544, −7.15403052753511678244160264322, −6.46804541261126352386271868940, −6.13990077712613125253657360619, −4.91703975611897177189637089883, −3.9749190450229034229667240461, −3.50975883075815070100942686785, −2.4951337601384053516657917435, −1.34186959618188371358078013345, 0.69038911700762011879791528580, 1.67587450820519932671169368717, 2.43256957976757885624502889761, 3.419548275325925973405676461808, 4.23372342753491470030604764740, 5.26734904850160693087326042565, 5.67990039131134288021730895409, 6.38484004721811714135494300172, 7.63373979667290282720477661557, 8.75307902687157603960151801507, 9.39919672802778913215001380122, 9.88023477597393676177363708091, 10.8879991292611747329867184632, 11.87612318377474052533126118694, 12.310736854938394152446376003759, 12.89407942577079003807094168597, 13.64319267820261537534982503909, 14.5543361877367783877984751058, 15.00919827895259436669503137340, 16.10585685135796446629059744198, 16.48879005552855206977389503680, 17.84296884017862417986822537393, 18.139939865854557429770681536166, 19.23523865245597201507111885629, 19.74590165594872353356905338347

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