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Properties

Label 1-2011-2011.28-r1-0-0
Degree $1$
Conductor $2011$
Sign $0.263 + 0.964i$
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  + (0.994 − 0.109i)2-s + (0.623 + 0.781i)3-s + (0.976 − 0.217i)4-s + (0.230 + 0.973i)5-s + (0.705 + 0.708i)6-s + (0.995 + 0.0905i)7-s + (0.946 − 0.322i)8-s + (−0.221 + 0.975i)9-s + (0.335 + 0.941i)10-s + (0.907 + 0.420i)11-s + (0.778 + 0.627i)12-s + (−0.191 − 0.981i)13-s + (0.999 − 0.0187i)14-s + (−0.616 + 0.787i)15-s + (0.905 − 0.423i)16-s + (−0.957 − 0.289i)17-s + ⋯
L(s)  = 1  + (0.994 − 0.109i)2-s + (0.623 + 0.781i)3-s + (0.976 − 0.217i)4-s + (0.230 + 0.973i)5-s + (0.705 + 0.708i)6-s + (0.995 + 0.0905i)7-s + (0.946 − 0.322i)8-s + (−0.221 + 0.975i)9-s + (0.335 + 0.941i)10-s + (0.907 + 0.420i)11-s + (0.778 + 0.627i)12-s + (−0.191 − 0.981i)13-s + (0.999 − 0.0187i)14-s + (−0.616 + 0.787i)15-s + (0.905 − 0.423i)16-s + (−0.957 − 0.289i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ 0.263 + 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.810695555 + 5.198408357i\)
\(L(\frac12)\) \(\approx\) \(6.810695555 + 5.198408357i\)
\(L(1)\) \(\approx\) \(2.933521506 + 1.125538294i\)
\(L(1)\) \(\approx\) \(2.933521506 + 1.125538294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.994 - 0.109i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.230 + 0.973i)T \)
7 \( 1 + (0.995 + 0.0905i)T \)
11 \( 1 + (0.907 + 0.420i)T \)
13 \( 1 + (-0.191 - 0.981i)T \)
17 \( 1 + (-0.957 - 0.289i)T \)
19 \( 1 + (0.929 + 0.369i)T \)
23 \( 1 + (-0.294 - 0.955i)T \)
29 \( 1 + (0.589 - 0.808i)T \)
31 \( 1 + (0.978 + 0.204i)T \)
37 \( 1 + (0.359 - 0.933i)T \)
41 \( 1 + (-0.958 + 0.286i)T \)
43 \( 1 + (0.985 + 0.168i)T \)
47 \( 1 + (0.489 + 0.872i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.405 + 0.914i)T \)
61 \( 1 + (0.129 - 0.991i)T \)
67 \( 1 + (0.526 + 0.849i)T \)
71 \( 1 + (0.169 + 0.985i)T \)
73 \( 1 + (0.110 + 0.993i)T \)
79 \( 1 + (-0.453 + 0.891i)T \)
83 \( 1 + (-0.981 + 0.189i)T \)
89 \( 1 + (-0.954 - 0.298i)T \)
97 \( 1 + (-0.960 - 0.277i)T \)
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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

−19.85428634494415377506929068992, −19.21018886064864503431645033631, −18.01491865262076954203722053710, −17.327381387238220499583079006, −16.75303535180887889006131185959, −15.76466778824083538033957742806, −15.04854291941994279914674748441, −14.16214529013481643130001071872, −13.724047963989402585352695922432, −13.32123018009749530467051213716, −12.11037655826109314962285209207, −11.87599797872588001645035806612, −11.18452261389006906893062750550, −9.78734324899597637992896418279, −8.81806653530678461214050513441, −8.37028580288843569027504769793, −7.39421119634802401610371622939, −6.7212339168617729266773107932, −5.90049554494835892320517126113, −4.959236178885811552299027166704, −4.27377301079528582178787187608, −3.47089310780871113362019348838, −2.25931462137121642103381820663, −1.61867287042019660206508254367, −0.94839570796267532505890042195, 1.21594337369365021344768577927, 2.51386183400136535105452982065, 2.60269567141970628001415190456, 3.88897870957869646826174258491, 4.361795053673975716148946220164, 5.26730191830909119009408951171, 6.05189871446699545631851793763, 7.05298104298311127931701278731, 7.74548938913121034302800355554, 8.61350763425367075203921960268, 9.82729254877515492080558615296, 10.280206488007075074290894904191, 11.14885296252933554661832129510, 11.61379969730543786107577079611, 12.601110347309961322396834104956, 13.69865327515056350128085133501, 14.22793426049231131711516129813, 14.57117871277040908549904117068, 15.47245724890847213595003454608, 15.6843734894706814653947118705, 16.96731489783788645424774517615, 17.6477563996931671706596217588, 18.53799989391371295807835998720, 19.56027860659660782163154001306, 20.11224865799199273627224096025

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