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Properties

Label 1-1012-1012.203-r0-0-0
Degree $1$
Conductor $1012$
Sign $0.0609 + 0.998i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (−0.774 − 0.633i)3-s + (−0.198 + 0.980i)5-s + (−0.0285 + 0.999i)7-s + (0.198 + 0.980i)9-s + (0.941 − 0.336i)13-s + (0.774 − 0.633i)15-s + (−0.696 + 0.717i)17-s + (0.897 − 0.441i)19-s + (0.654 − 0.755i)21-s + (−0.921 − 0.389i)25-s + (0.466 − 0.884i)27-s + (0.897 + 0.441i)29-s + (0.998 − 0.0570i)31-s + (−0.974 − 0.226i)35-s + (0.736 + 0.676i)37-s + ⋯
L(s)  = 1  + (−0.774 − 0.633i)3-s + (−0.198 + 0.980i)5-s + (−0.0285 + 0.999i)7-s + (0.198 + 0.980i)9-s + (0.941 − 0.336i)13-s + (0.774 − 0.633i)15-s + (−0.696 + 0.717i)17-s + (0.897 − 0.441i)19-s + (0.654 − 0.755i)21-s + (−0.921 − 0.389i)25-s + (0.466 − 0.884i)27-s + (0.897 + 0.441i)29-s + (0.998 − 0.0570i)31-s + (−0.974 − 0.226i)35-s + (0.736 + 0.676i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0609 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0609 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.0609 + 0.998i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ 0.0609 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7256458680 + 0.6826536053i\)
\(L(\frac12)\) \(\approx\) \(0.7256458680 + 0.6826536053i\)
\(L(1)\) \(\approx\) \(0.8175307669 + 0.1853948535i\)
\(L(1)\) \(\approx\) \(0.8175307669 + 0.1853948535i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.774 - 0.633i)T \)
5 \( 1 + (-0.198 + 0.980i)T \)
7 \( 1 + (-0.0285 + 0.999i)T \)
13 \( 1 + (0.941 - 0.336i)T \)
17 \( 1 + (-0.696 + 0.717i)T \)
19 \( 1 + (0.897 - 0.441i)T \)
29 \( 1 + (0.897 + 0.441i)T \)
31 \( 1 + (0.998 - 0.0570i)T \)
37 \( 1 + (0.736 + 0.676i)T \)
41 \( 1 + (-0.736 + 0.676i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.610 + 0.791i)T \)
59 \( 1 + (0.564 - 0.825i)T \)
61 \( 1 + (0.362 + 0.931i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (-0.974 + 0.226i)T \)
73 \( 1 + (-0.985 - 0.170i)T \)
79 \( 1 + (0.941 - 0.336i)T \)
83 \( 1 + (-0.870 + 0.491i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.870 + 0.491i)T \)
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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

−21.17768186644099332449792569334, −20.77850468986720604138509778243, −20.137642300761634742785244345608, −19.21136761834987100075537531479, −17.96202865483932741698247799613, −17.49833786074378828123321514401, −16.54747818587929714474848456009, −16.08579315222829653212342528020, −15.555995841749284328978974971019, −14.19007338856391688147583255341, −13.508327950702767675351975229425, −12.62029836858500565582680640174, −11.67125119089606714534381497138, −11.16862202436163453935458261297, −10.14983797568524034725210479965, −9.46509607484482239712974931786, −8.58996074669979090997580877153, −7.568802753452995695014783696546, −6.55295094018147259725913806364, −5.70824927484563295523630881210, −4.62869228520697936627308341471, −4.250759229619278814820246155826, −3.22160814053103593435609873821, −1.40236643704302367571083677326, −0.56280612858795318679774217758, 1.20142541896089242963942371786, 2.374619150066491929223709807669, 3.16454584313394962210379034383, 4.494268786941307341605478727695, 5.59329854836685314744792346625, 6.278583386483204598596254329667, 6.88745412005537334574206790936, 7.967324822528475854513653033061, 8.67051184636606290475851842434, 9.986304427059818889403922951656, 10.77073653987022639948164897392, 11.53121457270261550996040708153, 12.03755708508668068553337275763, 13.11136685237189292694508711160, 13.71675839028361876811971536449, 14.8299528949952940375677418194, 15.61107031750754135393341630922, 16.157333580766805969773355477143, 17.48362949257951233320870301199, 17.94598721137549135560623203342, 18.60800804441535701097460805273, 19.195779099751536412807728714407, 20.06520790183920867967390887220, 21.31665577153246459262368959826, 22.12254280445685130868167481219

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