Properties

Label 1-1011-1011.68-r0-0-0
Degree $1$
Conductor $1011$
Sign $0.999 + 0.0123i$
Analytic cond. $4.69506$
Root an. cond. $4.69506$
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  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.815 + 0.578i)5-s + (0.943 − 0.330i)7-s + (0.222 + 0.974i)8-s + (0.516 − 0.856i)10-s + (−0.998 − 0.0560i)11-s + (−0.733 + 0.680i)13-s + (−0.399 − 0.916i)14-s + (0.955 − 0.294i)16-s + (0.985 + 0.167i)17-s + (0.547 + 0.836i)19-s + (−0.892 − 0.450i)20-s + (0.0186 + 0.999i)22-s + (0.995 + 0.0933i)23-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.815 + 0.578i)5-s + (0.943 − 0.330i)7-s + (0.222 + 0.974i)8-s + (0.516 − 0.856i)10-s + (−0.998 − 0.0560i)11-s + (−0.733 + 0.680i)13-s + (−0.399 − 0.916i)14-s + (0.955 − 0.294i)16-s + (0.985 + 0.167i)17-s + (0.547 + 0.836i)19-s + (−0.892 − 0.450i)20-s + (0.0186 + 0.999i)22-s + (0.995 + 0.0933i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1011\)    =    \(3 \cdot 337\)
Sign: $0.999 + 0.0123i$
Analytic conductor: \(4.69506\)
Root analytic conductor: \(4.69506\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1011} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1011,\ (0:\ ),\ 0.999 + 0.0123i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.487850512 + 0.009157477704i\)
\(L(\frac12)\) \(\approx\) \(1.487850512 + 0.009157477704i\)
\(L(1)\) \(\approx\) \(1.086155503 - 0.2573285411i\)
\(L(1)\) \(\approx\) \(1.086155503 - 0.2573285411i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
337 \( 1 \)
good2 \( 1 + (-0.0747 - 0.997i)T \)
5 \( 1 + (0.815 + 0.578i)T \)
7 \( 1 + (0.943 - 0.330i)T \)
11 \( 1 + (-0.998 - 0.0560i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (0.985 + 0.167i)T \)
19 \( 1 + (0.547 + 0.836i)T \)
23 \( 1 + (0.995 + 0.0933i)T \)
29 \( 1 + (-0.908 + 0.416i)T \)
31 \( 1 + (-0.978 + 0.204i)T \)
37 \( 1 + (0.149 - 0.988i)T \)
41 \( 1 + (-0.997 + 0.0747i)T \)
43 \( 1 + (0.111 + 0.993i)T \)
47 \( 1 + (0.111 - 0.993i)T \)
53 \( 1 + (-0.416 + 0.908i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + (0.836 + 0.547i)T \)
67 \( 1 + (0.638 + 0.770i)T \)
71 \( 1 + (-0.547 + 0.836i)T \)
73 \( 1 + (0.450 - 0.892i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + (0.720 + 0.693i)T \)
89 \( 1 + (0.416 + 0.908i)T \)
97 \( 1 + (-0.578 - 0.815i)T \)
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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

−21.786250447625625377608335389739, −20.862323397602947606942830141249, −20.31715373952351076923607886690, −18.866791112035135879739702606204, −18.35590977518763723897151259070, −17.435567275534632385318834011270, −17.142315487996895754578438861711, −16.15145723920273706719832461528, −15.27071198019477300739163704681, −14.679476310054344390488143609519, −13.80467707502157951319247552950, −13.05433856233127417216337609749, −12.41082041221839813552705574670, −11.13034066382354018889736931264, −10.05819854125411056768779955049, −9.44738072417054445726102796865, −8.50418362504346635328527401083, −7.81153397342922223164432621159, −7.05847666593195890657846745242, −5.71900448425123852280049346370, −5.19275092438400422750440103848, −4.785585390076758316977352400033, −3.17807442185348037980321978528, −1.96097008648675004240142283186, −0.703182094234339492611418243002, 1.28955502817715027302973917522, 2.04613977295652655513406220063, 2.96369447091216662334871842589, 3.933145960358764912422245899839, 5.20911618821730625012156869242, 5.49847285128219760449155155231, 7.185580757289365111253829066232, 7.80527141884518908138345905468, 8.93261608477774274291210897813, 9.783113056882954444370997099167, 10.46297878134471280383570497172, 11.086344780828860047447591682, 11.9432134656300704369222638913, 12.894361243995009657317747312211, 13.61747116214044822011435875667, 14.5602822794515218087820487800, 14.71554155716991121058569425559, 16.49767617721337121636942591691, 17.1060419330590095953616619246, 18.00847518868628916797242786498, 18.4987014512291054329586116287, 19.15153221891947929470605719584, 20.30810059366879987540172217289, 20.93895185711636455464014972904, 21.44860992157493919671632874785

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