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Properties

Label 1-1012-1012.367-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.999 + 0.0237i$
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  + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s − 21-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)35-s + (0.809 − 0.587i)37-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)9-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s − 21-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.809 + 0.587i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)35-s + (0.809 − 0.587i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.999 + 0.0237i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.999 + 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01376162614 + 1.157653190i\)
\(L(\frac12)\) \(\approx\) \(0.01376162614 + 1.157653190i\)
\(L(1)\) \(\approx\) \(0.8601402008 + 0.6262739596i\)
\(L(1)\) \(\approx\) \(0.8601402008 + 0.6262739596i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−20.771950425828391047008192757231, −20.414620843986342171466571484152, −19.78117318944287204682892307505, −19.05757754229080827135207624088, −18.245342980029691052450039701, −17.28587457694445000847396837085, −16.49727667963877292197393337173, −15.69023894815017241440276115639, −14.98783842239923564459564289068, −13.84383115101151027615253388792, −13.25706113624805034808517687013, −12.677298998752690565094835529504, −11.95999221428604504267209631266, −10.73251915164749094947433420306, −9.71263059836572019568904816815, −9.05205048916755750551851429314, −8.17259876990109698702899532347, −7.53253219020747386886627291962, −6.60144517243048251163321501476, −5.63374631123120701311046176045, −4.37953750013201368355357677793, −3.60017128891548094111888493309, −2.69687359784043253837620722771, −1.433413880593750765140508855357, −0.42771247290845280546004978214, 2.03490017128544418141618589838, 2.633203240095589684751051823650, 3.74447335697168957315278524032, 4.163958287170833487205045529216, 5.64467213112217923864267258558, 6.55857601874509991047636368232, 7.33416703611903158388197302539, 8.43406564482580532790866033023, 9.11281681745396072927566903073, 9.87892508176047108569331713902, 10.80037545171523619737853099783, 11.390105924019558296850407197731, 12.67027451663673186085222687148, 13.35074798174108016088084835663, 14.36810855272347183587666106496, 14.92699127030476736695853765732, 15.606929832733869380060712400602, 16.27043555535227777025799332558, 17.23645038460183406264476650085, 18.473499391297268470425501129523, 19.070103712540999181102063804236, 19.473324673934289549899868469843, 20.42028655780156445771009392920, 21.43370988407375170442720881797, 21.95534905720973893481721353070

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