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Properties

Label 1-1012-1012.103-r0-0-0
Degree $1$
Conductor $1012$
Sign $0.494 - 0.868i$
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  + (−0.610 − 0.791i)3-s + (0.254 + 0.967i)5-s + (−0.466 − 0.884i)7-s + (−0.254 + 0.967i)9-s + (0.897 − 0.441i)13-s + (0.610 − 0.791i)15-s + (−0.516 − 0.856i)17-s + (0.0855 + 0.996i)19-s + (−0.415 + 0.909i)21-s + (−0.870 + 0.491i)25-s + (0.921 − 0.389i)27-s + (0.0855 − 0.996i)29-s + (0.564 + 0.825i)31-s + (0.736 − 0.676i)35-s + (0.998 − 0.0570i)37-s + ⋯
L(s)  = 1  + (−0.610 − 0.791i)3-s + (0.254 + 0.967i)5-s + (−0.466 − 0.884i)7-s + (−0.254 + 0.967i)9-s + (0.897 − 0.441i)13-s + (0.610 − 0.791i)15-s + (−0.516 − 0.856i)17-s + (0.0855 + 0.996i)19-s + (−0.415 + 0.909i)21-s + (−0.870 + 0.491i)25-s + (0.921 − 0.389i)27-s + (0.0855 − 0.996i)29-s + (0.564 + 0.825i)31-s + (0.736 − 0.676i)35-s + (0.998 − 0.0570i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.494 - 0.868i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ 0.494 - 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9439884561 - 0.5487201517i\)
\(L(\frac12)\) \(\approx\) \(0.9439884561 - 0.5487201517i\)
\(L(1)\) \(\approx\) \(0.8640126818 - 0.1999221829i\)
\(L(1)\) \(\approx\) \(0.8640126818 - 0.1999221829i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.610 - 0.791i)T \)
5 \( 1 + (0.254 + 0.967i)T \)
7 \( 1 + (-0.466 - 0.884i)T \)
13 \( 1 + (0.897 - 0.441i)T \)
17 \( 1 + (-0.516 - 0.856i)T \)
19 \( 1 + (0.0855 + 0.996i)T \)
29 \( 1 + (0.0855 - 0.996i)T \)
31 \( 1 + (0.564 + 0.825i)T \)
37 \( 1 + (0.998 - 0.0570i)T \)
41 \( 1 + (-0.998 - 0.0570i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.985 - 0.170i)T \)
59 \( 1 + (-0.696 - 0.717i)T \)
61 \( 1 + (0.0285 - 0.999i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.736 + 0.676i)T \)
73 \( 1 + (0.974 + 0.226i)T \)
79 \( 1 + (0.897 - 0.441i)T \)
83 \( 1 + (0.774 - 0.633i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (-0.774 - 0.633i)T \)
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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

−21.719082635580370650722769497830, −21.2000098098223760120042356100, −20.28512077986656306995705736240, −19.610968286593146316248465353458, −18.42056746752285389563177326368, −17.8127973370525761906480795003, −16.7796919595279355860074142906, −16.421789232078254325453792828309, −15.43469371765519981079656070596, −15.11422653686211162494554285101, −13.64692902752889162144802352369, −12.97981621878788105669474216313, −12.09445720629917960992093910921, −11.44856002552172026300337402564, −10.512438788154610405235154810138, −9.55156312396454828843904043164, −8.9313885513737424150206792204, −8.37737895629324607292976620385, −6.68117800054533869570005605901, −6.02205034436598606529683125687, −5.24069442108823995604533685884, −4.42701524230152848068292615680, −3.53719963938204162700535144164, −2.26252264504476992474930029048, −0.95460372272198066109566265795, 0.64546467493638193832867900597, 1.83295450355457424547616900118, 2.9288023733281824741219337498, 3.83997165184425838650465715244, 5.11931038233943594900915569671, 6.227118863749594520104080554878, 6.58939953963516760461408556802, 7.50153817134331372235450972317, 8.20882240608667656808203190706, 9.65445185567324749565544256108, 10.4370362884871370092876271006, 11.04523863125710111529392939937, 11.842827583279459592262109109326, 12.84393068841194331587153173610, 13.80966887122100214589654004514, 13.86002712364701031854217639885, 15.252750858318554548891486869163, 16.116750329673116417248442098862, 16.93643660680157565308625026386, 17.68629516623577551266344309025, 18.40846635042252295374858771957, 18.903154306015415239417069348626, 19.85806948841334624395030534888, 20.596031557194085740278396349347, 21.71897111116202434117433219496

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