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Properties

Label 1-2011-2011.66-r1-0-0
Degree $1$
Conductor $2011$
Sign $0.0517 + 0.998i$
Analytic cond. $216.111$
Root an. cond. $216.111$
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  + (−0.752 + 0.658i)2-s + (−0.645 − 0.763i)3-s + (0.132 − 0.991i)4-s + (−0.694 − 0.719i)5-s + (0.988 + 0.149i)6-s + (−0.886 + 0.463i)7-s + (0.553 + 0.833i)8-s + (−0.166 + 0.986i)9-s + (0.996 + 0.0843i)10-s + (0.893 − 0.449i)11-s + (−0.842 + 0.538i)12-s + (0.965 + 0.259i)13-s + (0.362 − 0.932i)14-s + (−0.101 + 0.994i)15-s + (−0.964 − 0.262i)16-s + (−0.999 + 0.0437i)17-s + ⋯
L(s)  = 1  + (−0.752 + 0.658i)2-s + (−0.645 − 0.763i)3-s + (0.132 − 0.991i)4-s + (−0.694 − 0.719i)5-s + (0.988 + 0.149i)6-s + (−0.886 + 0.463i)7-s + (0.553 + 0.833i)8-s + (−0.166 + 0.986i)9-s + (0.996 + 0.0843i)10-s + (0.893 − 0.449i)11-s + (−0.842 + 0.538i)12-s + (0.965 + 0.259i)13-s + (0.362 − 0.932i)14-s + (−0.101 + 0.994i)15-s + (−0.964 − 0.262i)16-s + (−0.999 + 0.0437i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0517 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0517 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $0.0517 + 0.998i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ 0.0517 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4520132516 + 0.4292142113i\)
\(L(\frac12)\) \(\approx\) \(0.4520132516 + 0.4292142113i\)
\(L(1)\) \(\approx\) \(0.5241760793 + 0.02566001393i\)
\(L(1)\) \(\approx\) \(0.5241760793 + 0.02566001393i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.752 + 0.658i)T \)
3 \( 1 + (-0.645 - 0.763i)T \)
5 \( 1 + (-0.694 - 0.719i)T \)
7 \( 1 + (-0.886 + 0.463i)T \)
11 \( 1 + (0.893 - 0.449i)T \)
13 \( 1 + (0.965 + 0.259i)T \)
17 \( 1 + (-0.999 + 0.0437i)T \)
19 \( 1 + (-0.601 + 0.798i)T \)
23 \( 1 + (0.967 - 0.253i)T \)
29 \( 1 + (0.863 - 0.504i)T \)
31 \( 1 + (0.803 + 0.595i)T \)
37 \( 1 + (0.0546 + 0.998i)T \)
41 \( 1 + (0.970 + 0.241i)T \)
43 \( 1 + (-0.191 - 0.981i)T \)
47 \( 1 + (-0.273 + 0.961i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.00468 + 0.999i)T \)
61 \( 1 + (0.433 - 0.901i)T \)
67 \( 1 + (0.578 - 0.815i)T \)
71 \( 1 + (-0.0920 + 0.995i)T \)
73 \( 1 + (-0.682 + 0.730i)T \)
79 \( 1 + (-0.261 + 0.965i)T \)
83 \( 1 + (0.934 + 0.354i)T \)
89 \( 1 + (0.849 + 0.528i)T \)
97 \( 1 + (-0.664 + 0.747i)T \)
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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

−19.61900052097357858480272088671, −19.0257516812165519911678653348, −17.93057797054362921462572781079, −17.59482057779940391386950185137, −16.7239681975137791885482516741, −15.99794637277600591011126902512, −15.53146101237455892519505728995, −14.70955627703058593896974732274, −13.46452649769091302009220749830, −12.75915580187209360977615197669, −11.849815537048978072240517805642, −11.24033493638336103897903106930, −10.6997598280810949172769916023, −10.1189111944732691010706977874, −9.15289177810031581894142237725, −8.74563687123642548127650977447, −7.44401288851303165712644633032, −6.66536541561537028984221900950, −6.307888811821602643931298596453, −4.58669432083271707771108321624, −4.0162765312539337662541318304, −3.3587062534958632176161169813, −2.53790959296435343178794213267, −0.98217936968783684179239596014, −0.26987316192837048974374341295, 0.7726255815772196625199282084, 1.32403230005237185900374013009, 2.53298668410417770086984796837, 3.92593837238981958583534720003, 4.84726762949451011441273123268, 5.82146217818660041356062221331, 6.51627607585715934936691598485, 6.82662363094886921827757821103, 8.09291236539254987479655379406, 8.579256999990054009486929657649, 9.14641782728533920256614682954, 10.257557687546227858625774778117, 11.157894753775543885219788894417, 11.70513529345961131518741721438, 12.532710241868723390997830653075, 13.30116071398134070075745584268, 14.00880415773696312445533860715, 15.15649967730302668330812017731, 15.86197418108673937186073436459, 16.38378544520938914062073481587, 16.97829841483531522790579473018, 17.625661739130401871748621337409, 18.5971915264024153590628910999, 19.15630066648442305591773949617, 19.46299134008295693440931586182

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