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Properties

Label 1-1012-1012.467-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.185 + 0.982i$
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  + (−0.993 − 0.113i)3-s + (−0.974 + 0.226i)5-s + (−0.254 − 0.967i)7-s + (0.974 + 0.226i)9-s + (−0.998 + 0.0570i)13-s + (0.993 − 0.113i)15-s + (−0.610 − 0.791i)17-s + (−0.564 − 0.825i)19-s + (0.142 + 0.989i)21-s + (0.897 − 0.441i)25-s + (−0.941 − 0.336i)27-s + (−0.564 + 0.825i)29-s + (0.870 + 0.491i)31-s + (0.466 + 0.884i)35-s + (0.921 − 0.389i)37-s + ⋯
L(s)  = 1  + (−0.993 − 0.113i)3-s + (−0.974 + 0.226i)5-s + (−0.254 − 0.967i)7-s + (0.974 + 0.226i)9-s + (−0.998 + 0.0570i)13-s + (0.993 − 0.113i)15-s + (−0.610 − 0.791i)17-s + (−0.564 − 0.825i)19-s + (0.142 + 0.989i)21-s + (0.897 − 0.441i)25-s + (−0.941 − 0.336i)27-s + (−0.564 + 0.825i)29-s + (0.870 + 0.491i)31-s + (0.466 + 0.884i)35-s + (0.921 − 0.389i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.185 + 0.982i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.185 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09979873528 + 0.1203641968i\)
\(L(\frac12)\) \(\approx\) \(0.09979873528 + 0.1203641968i\)
\(L(1)\) \(\approx\) \(0.5036957725 - 0.07175706421i\)
\(L(1)\) \(\approx\) \(0.5036957725 - 0.07175706421i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.993 - 0.113i)T \)
5 \( 1 + (-0.974 + 0.226i)T \)
7 \( 1 + (-0.254 - 0.967i)T \)
13 \( 1 + (-0.998 + 0.0570i)T \)
17 \( 1 + (-0.610 - 0.791i)T \)
19 \( 1 + (-0.564 - 0.825i)T \)
29 \( 1 + (-0.564 + 0.825i)T \)
31 \( 1 + (0.870 + 0.491i)T \)
37 \( 1 + (0.921 - 0.389i)T \)
41 \( 1 + (-0.921 - 0.389i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.362 - 0.931i)T \)
59 \( 1 + (-0.774 + 0.633i)T \)
61 \( 1 + (-0.198 + 0.980i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.466 - 0.884i)T \)
73 \( 1 + (-0.0285 + 0.999i)T \)
79 \( 1 + (-0.998 + 0.0570i)T \)
83 \( 1 + (0.0855 + 0.996i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (-0.0855 + 0.996i)T \)
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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

−21.62950165157112190337542954655, −20.73407207869424105807410102203, −19.67218717566954219297170651139, −18.98611147903041142225155122752, −18.4140479079576714273057227936, −17.26407110253131345649277030447, −16.81606856176025417397500244964, −15.830453042886534570124764844370, −15.29370207487617955078895721995, −14.653355421459929749597553818621, −13.00670123221280053427792345704, −12.56581868902829325074227673052, −11.78245197032259668811752764207, −11.2593454057230261993416475644, −10.19343357301756904986345048396, −9.418467048369520928851657639791, −8.31828961810558487680220985239, −7.58732743228394951644740747727, −6.44034692609546726567905107897, −5.84529865305443780350283187454, −4.71698879185869161245050928809, −4.18131397853327079940689300632, −2.905626463839249149678617236655, −1.66243522810603969620157649523, −0.10421972814093119820278273883, 0.84240016061498049541812604297, 2.39786558599806786743108139370, 3.6523651432272216818043433075, 4.52795054462439809064249596289, 5.11704367110025024869134296974, 6.59221306906017335031692159311, 7.04349249237373250561880134241, 7.68600323575093396077696673500, 8.952744862770519015765476542769, 10.0666001643845020273637890069, 10.74582782642933786027429274876, 11.446500716243302920775550906080, 12.16353350180881383307240437240, 12.98017387820246256767324072058, 13.81949118808825915313260263338, 14.943414696465072565982440332030, 15.659988580874486333347791117378, 16.48650955483817929567374743292, 17.032134561545931618957875582276, 17.885462202395743842037216247993, 18.67510123476557326964071984954, 19.64562432529448373776475223187, 19.97943137667729483096182370251, 21.16870356176360471514067494276, 22.155791154701081724246377948041

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