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Properties

Label 1-2001-2001.485-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.734 + 0.678i$
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  + (−0.242 + 0.970i)2-s + (−0.882 − 0.470i)4-s + (0.947 − 0.320i)5-s + (0.996 + 0.0815i)7-s + (0.670 − 0.742i)8-s + (0.0815 + 0.996i)10-s + (−0.999 + 0.0203i)11-s + (0.301 + 0.953i)13-s + (−0.320 + 0.947i)14-s + (0.557 + 0.830i)16-s + (0.755 − 0.654i)17-s + (0.470 − 0.882i)19-s + (−0.986 − 0.162i)20-s + (0.222 − 0.974i)22-s + (0.794 − 0.607i)25-s + (−0.998 + 0.0611i)26-s + ⋯
L(s)  = 1  + (−0.242 + 0.970i)2-s + (−0.882 − 0.470i)4-s + (0.947 − 0.320i)5-s + (0.996 + 0.0815i)7-s + (0.670 − 0.742i)8-s + (0.0815 + 0.996i)10-s + (−0.999 + 0.0203i)11-s + (0.301 + 0.953i)13-s + (−0.320 + 0.947i)14-s + (0.557 + 0.830i)16-s + (0.755 − 0.654i)17-s + (0.470 − 0.882i)19-s + (−0.986 − 0.162i)20-s + (0.222 − 0.974i)22-s + (0.794 − 0.607i)25-s + (−0.998 + 0.0611i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.734 + 0.678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.720633161 + 0.6725471135i\)
\(L(\frac12)\) \(\approx\) \(1.720633161 + 0.6725471135i\)
\(L(1)\) \(\approx\) \(1.117752746 + 0.4004114635i\)
\(L(1)\) \(\approx\) \(1.117752746 + 0.4004114635i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.242 + 0.970i)T \)
5 \( 1 + (0.947 - 0.320i)T \)
7 \( 1 + (0.996 + 0.0815i)T \)
11 \( 1 + (-0.999 + 0.0203i)T \)
13 \( 1 + (0.301 + 0.953i)T \)
17 \( 1 + (0.755 - 0.654i)T \)
19 \( 1 + (0.470 - 0.882i)T \)
31 \( 1 + (0.574 + 0.818i)T \)
37 \( 1 + (-0.122 - 0.992i)T \)
41 \( 1 + (0.540 - 0.841i)T \)
43 \( 1 + (-0.574 + 0.818i)T \)
47 \( 1 + (0.433 + 0.900i)T \)
53 \( 1 + (0.862 + 0.505i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (-0.872 - 0.488i)T \)
67 \( 1 + (-0.0203 + 0.999i)T \)
71 \( 1 + (-0.768 - 0.639i)T \)
73 \( 1 + (0.0407 - 0.999i)T \)
79 \( 1 + (-0.830 - 0.557i)T \)
83 \( 1 + (0.714 + 0.699i)T \)
89 \( 1 + (-0.925 - 0.377i)T \)
97 \( 1 + (0.940 + 0.339i)T \)
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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

−20.1398673318356106770494947701, −18.92363780049706969749745738963, −18.36196886672606607892289894383, −17.96717580644768013144048742373, −17.17118645770913979185024132892, −16.60342979984728459910136518882, −15.240466194446920714553655681963, −14.63456204205929506650805849329, −13.71006009308984950624459487857, −13.32006623811287171880085513966, −12.443698717890250470572347255214, −11.65936937098168101868666123289, −10.77227152225871258190043916502, −10.23783933112518771049747081269, −9.828440175847625176037844252688, −8.54385247424424290039015779271, −8.10254009750472361559184444147, −7.31359308651117723080147680345, −5.75321348644237172557595495399, −5.446836458202788264226318392135, −4.43305603077784279121083548952, −3.35640782710509396997824570558, −2.62767855174999883842493562433, −1.7482094055505704875672296038, −1.00160018074369994630961966953, 0.88356116653506908516599526350, 1.76600493745091613052462498225, 2.819993841630035102067092308182, 4.29653423385845407874280934723, 5.036437091936780805501178207971, 5.46270275861127184600232932670, 6.382184209212052958544841877534, 7.294894785982623665954784252592, 7.924209354562214961876406266511, 8.89173523008082456985700560094, 9.28655346133321062332437818794, 10.251972328947654567689076020182, 10.922649161955598658506320836644, 11.99713816617186542135297659713, 12.94980366008036434392569346339, 13.80473239230641597589203106861, 14.07627193123412661005743336329, 14.887146280630131621094713792063, 15.90453821736098025697390266608, 16.27586582316703814173480957781, 17.26659500956427543739266604893, 17.74972128832137431835070582007, 18.33403658938046126495894066650, 18.95096623284075071894782171660, 20.05483230902307728298619088491

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