↑

Properties

Label 1-2011-2011.7-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.717 - 0.696i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0390 + 0.999i)2-s + (0.805 − 0.592i)3-s + (−0.996 − 0.0780i)4-s + (−0.872 − 0.487i)5-s + (0.560 + 0.827i)6-s + (−0.999 − 0.0125i)7-s + (0.116 − 0.993i)8-s + (0.297 − 0.954i)9-s + (0.521 − 0.853i)10-s + (−0.113 + 0.993i)11-s + (−0.849 + 0.528i)12-s + (−0.135 + 0.990i)13-s + (0.0515 − 0.998i)14-s + (−0.992 + 0.124i)15-s + (0.987 + 0.155i)16-s + (0.959 − 0.280i)17-s + ⋯
L(s)  = 1  + (−0.0390 + 0.999i)2-s + (0.805 − 0.592i)3-s + (−0.996 − 0.0780i)4-s + (−0.872 − 0.487i)5-s + (0.560 + 0.827i)6-s + (−0.999 − 0.0125i)7-s + (0.116 − 0.993i)8-s + (0.297 − 0.954i)9-s + (0.521 − 0.853i)10-s + (−0.113 + 0.993i)11-s + (−0.849 + 0.528i)12-s + (−0.135 + 0.990i)13-s + (0.0515 − 0.998i)14-s + (−0.992 + 0.124i)15-s + (0.987 + 0.155i)16-s + (0.959 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.717 - 0.696i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.717 - 0.696i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03419766851 - 0.08436469412i\)
\(L(\frac12)\) \(\approx\) \(0.03419766851 - 0.08436469412i\)
\(L(1)\) \(\approx\) \(0.8149601670 + 0.2068183596i\)
\(L(1)\) \(\approx\) \(0.8149601670 + 0.2068183596i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.0390 + 0.999i)T \)
3 \( 1 + (0.805 - 0.592i)T \)
5 \( 1 + (-0.872 - 0.487i)T \)
7 \( 1 + (-0.999 - 0.0125i)T \)
11 \( 1 + (-0.113 + 0.993i)T \)
13 \( 1 + (-0.135 + 0.990i)T \)
17 \( 1 + (0.959 - 0.280i)T \)
19 \( 1 + (-0.964 - 0.265i)T \)
23 \( 1 + (0.0951 + 0.995i)T \)
29 \( 1 + (0.288 + 0.957i)T \)
31 \( 1 + (-0.537 - 0.843i)T \)
37 \( 1 + (0.416 - 0.909i)T \)
41 \( 1 + (0.0140 - 0.999i)T \)
43 \( 1 + (0.447 + 0.894i)T \)
47 \( 1 + (-0.529 - 0.848i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.685 - 0.728i)T \)
61 \( 1 + (0.847 + 0.530i)T \)
67 \( 1 + (0.0858 - 0.996i)T \)
71 \( 1 + (-0.184 - 0.982i)T \)
73 \( 1 + (0.573 + 0.819i)T \)
79 \( 1 + (-0.596 - 0.802i)T \)
83 \( 1 + (-0.707 + 0.706i)T \)
89 \( 1 + (0.889 + 0.457i)T \)
97 \( 1 + (0.698 + 0.715i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.95375711383885683167217142639, −19.29213957495778001238012351565, −18.99554251233944946940785754768, −18.33134884833322791041365646824, −17.01246421111568523403322721814, −16.3414899381076483952871032537, −15.60135404376624810606711370495, −14.71339998252464790290943100929, −14.29682703783716059312805078850, −13.18218327930279469495147865398, −12.79976094186631700408330756499, −11.86540613534652236131210852437, −10.95606691031942500962188228893, −10.252575904233845350594236501782, −9.990623239779240580822958574445, −8.72402318689039511577479540605, −8.34360235488774725368480624076, −7.59897949325382257525149158011, −6.32384431708113667294004641991, −5.31977805326265286378159535349, −4.225699035681131135329935248129, −3.621170972603458664152968875287, −2.98460852222627016748430177758, −2.5071742086547611434139288221, −0.91556507079137197689654548234, 0.019686959198822284381535657317, 1.01630455908328596265545313419, 2.21616103092738552974954885477, 3.52567816932283035565310317770, 3.97480395082346735281608791007, 4.92502115081018102625173288253, 5.99379661569188451901762581345, 7.02167641552790937671286385290, 7.247178225188567590827037371957, 8.04004836951947187652185626525, 9.027601954025911637858333282243, 9.313052022766649473397920086470, 10.17733596474920600667445003125, 11.651175997819293515187072080510, 12.60871357761824384618099169687, 12.77000117646062464936045291205, 13.68724112171293526773795548250, 14.54341517050387892451897089771, 15.0907311936183946588714900669, 15.80164191314059776334389872703, 16.46510758193381332284146377004, 17.13542793343757976614915237815, 18.11711156030899014628117246017, 18.82376491041886652995561158371, 19.4366722577932285289056444032

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