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Properties

Label 1-2011-2011.814-r1-0-0
Degree $1$
Conductor $2011$
Sign $0.685 - 0.728i$
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  + (0.0546 − 0.998i)2-s + (0.990 − 0.134i)3-s + (−0.994 − 0.109i)4-s + (0.832 + 0.554i)5-s + (−0.0796 − 0.996i)6-s + (−0.550 + 0.834i)7-s + (−0.163 + 0.986i)8-s + (0.964 − 0.265i)9-s + (0.599 − 0.800i)10-s + (0.399 − 0.916i)11-s + (−0.999 + 0.0250i)12-s + (−0.968 − 0.250i)13-s + (0.803 + 0.595i)14-s + (0.899 + 0.437i)15-s + (0.976 + 0.217i)16-s + (−0.886 − 0.463i)17-s + ⋯
L(s)  = 1  + (0.0546 − 0.998i)2-s + (0.990 − 0.134i)3-s + (−0.994 − 0.109i)4-s + (0.832 + 0.554i)5-s + (−0.0796 − 0.996i)6-s + (−0.550 + 0.834i)7-s + (−0.163 + 0.986i)8-s + (0.964 − 0.265i)9-s + (0.599 − 0.800i)10-s + (0.399 − 0.916i)11-s + (−0.999 + 0.0250i)12-s + (−0.968 − 0.250i)13-s + (0.803 + 0.595i)14-s + (0.899 + 0.437i)15-s + (0.976 + 0.217i)16-s + (−0.886 − 0.463i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $0.685 - 0.728i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (814, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ 0.685 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.094443182 - 1.336556316i\)
\(L(\frac12)\) \(\approx\) \(3.094443182 - 1.336556316i\)
\(L(1)\) \(\approx\) \(1.427406758 - 0.5730651889i\)
\(L(1)\) \(\approx\) \(1.427406758 - 0.5730651889i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.0546 - 0.998i)T \)
3 \( 1 + (0.990 - 0.134i)T \)
5 \( 1 + (0.832 + 0.554i)T \)
7 \( 1 + (-0.550 + 0.834i)T \)
11 \( 1 + (0.399 - 0.916i)T \)
13 \( 1 + (-0.968 - 0.250i)T \)
17 \( 1 + (-0.886 - 0.463i)T \)
19 \( 1 + (0.729 + 0.684i)T \)
23 \( 1 + (-0.948 + 0.316i)T \)
29 \( 1 + (0.890 - 0.454i)T \)
31 \( 1 + (0.744 - 0.667i)T \)
37 \( 1 + (-0.566 + 0.824i)T \)
41 \( 1 + (-0.896 - 0.443i)T \)
43 \( 1 + (0.855 + 0.517i)T \)
47 \( 1 + (0.0983 - 0.995i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.0515 + 0.998i)T \)
61 \( 1 + (0.975 - 0.220i)T \)
67 \( 1 + (-0.486 - 0.873i)T \)
71 \( 1 + (0.849 + 0.528i)T \)
73 \( 1 + (0.914 - 0.403i)T \)
79 \( 1 + (0.233 - 0.972i)T \)
83 \( 1 + (-0.662 + 0.749i)T \)
89 \( 1 + (0.986 + 0.161i)T \)
97 \( 1 + (0.990 - 0.140i)T \)
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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

−19.805265515191278636220936433781, −19.34457788842291952895750109941, −18.07615321378343317634331909789, −17.5502750099268206253720579567, −16.975470189181518962926574147950, −16.00442992913435373782197169767, −15.65141411306408217179371018323, −14.508153375536084072006634089588, −14.14240213028997022904247863463, −13.497432866597125881916579381172, −12.78857761619836557526571356024, −12.221868673591190564153041877640, −10.38571663031711677913790565958, −9.902274028270133919117548287974, −9.310802245924952592926588347, −8.66481749656327712268663980199, −7.7741166095121670366574579477, −6.85588729949831859306986387603, −6.62211353773422609042953991408, −5.21041949079254914185817003447, −4.5257656653685072142209873604, −3.96168049234008704468659188577, −2.763916770140648978057631959879, −1.770672852523140452495055865, −0.66094297025408844662550984721, 0.733311065076588986283203016806, 1.92410506100812931635260626637, 2.512406543965708762521256997583, 3.08752005790189323145375295179, 3.87475145710331182430286619582, 5.04257920946381933628099451558, 5.91774169170621296639015073414, 6.728446809165444579642748266316, 7.90719154048299479899080417092, 8.68251502153540012381892878608, 9.40801729840167545658035556464, 9.88110924500603146810082233724, 10.52116676658614611318851618300, 11.82138673351669028559656980965, 12.121297532333317875777717811376, 13.3016571025162475319228483362, 13.64958030051599267188917152110, 14.26458305935559385799255074180, 15.04832120371117254744946352079, 15.80494728847768083113967128550, 16.981737348562875325430603428256, 17.86935721641117028872737778070, 18.47166160400633655133199037738, 19.015092727042469088544713405783, 19.6566215413831864936087055439

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