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Properties

Label 1-1012-1012.383-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.590 + 0.806i$
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  + (−0.516 − 0.856i)3-s + (0.466 + 0.884i)5-s + (−0.736 + 0.676i)7-s + (−0.466 + 0.884i)9-s + (−0.870 + 0.491i)13-s + (0.516 − 0.856i)15-s + (0.362 − 0.931i)17-s + (0.774 − 0.633i)19-s + (0.959 + 0.281i)21-s + (−0.564 + 0.825i)25-s + (0.998 − 0.0570i)27-s + (0.774 + 0.633i)29-s + (−0.0855 − 0.996i)31-s + (−0.941 − 0.336i)35-s + (−0.897 + 0.441i)37-s + ⋯
L(s)  = 1  + (−0.516 − 0.856i)3-s + (0.466 + 0.884i)5-s + (−0.736 + 0.676i)7-s + (−0.466 + 0.884i)9-s + (−0.870 + 0.491i)13-s + (0.516 − 0.856i)15-s + (0.362 − 0.931i)17-s + (0.774 − 0.633i)19-s + (0.959 + 0.281i)21-s + (−0.564 + 0.825i)25-s + (0.998 − 0.0570i)27-s + (0.774 + 0.633i)29-s + (−0.0855 − 0.996i)31-s + (−0.941 − 0.336i)35-s + (−0.897 + 0.441i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.590 + 0.806i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.590 + 0.806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2454482640 + 0.4838070070i\)
\(L(\frac12)\) \(\approx\) \(0.2454482640 + 0.4838070070i\)
\(L(1)\) \(\approx\) \(0.7396880930 + 0.07446928279i\)
\(L(1)\) \(\approx\) \(0.7396880930 + 0.07446928279i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.516 - 0.856i)T \)
5 \( 1 + (0.466 + 0.884i)T \)
7 \( 1 + (-0.736 + 0.676i)T \)
13 \( 1 + (-0.870 + 0.491i)T \)
17 \( 1 + (0.362 - 0.931i)T \)
19 \( 1 + (0.774 - 0.633i)T \)
29 \( 1 + (0.774 + 0.633i)T \)
31 \( 1 + (-0.0855 - 0.996i)T \)
37 \( 1 + (-0.897 + 0.441i)T \)
41 \( 1 + (0.897 + 0.441i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.198 + 0.980i)T \)
59 \( 1 + (-0.993 - 0.113i)T \)
61 \( 1 + (-0.974 - 0.226i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.941 + 0.336i)T \)
73 \( 1 + (-0.254 + 0.967i)T \)
79 \( 1 + (-0.870 + 0.491i)T \)
83 \( 1 + (0.696 + 0.717i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.696 + 0.717i)T \)
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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

−21.36941103987682280008956814695, −20.64304857871888669489606557318, −19.938769672873874592441363636221, −19.26448017324084344055199907013, −17.796334210199961361144628239833, −17.36002893071935945266485033501, −16.54155224915031475682400311673, −16.138203393573844468286380939939, −15.18897006279364961385949598845, −14.24256980358052776407943628247, −13.4067708136213396077544486709, −12.36316223616493624953073426391, −12.04916778093845043764002138226, −10.51687620699405383136680958693, −10.198632537172824373507273797271, −9.44117572399880785665933959342, −8.57700302067169311425247215644, −7.4823774855325096290006837929, −6.33650546117255007178836482829, −5.57782170663152650637519154796, −4.82072254619544124502636330600, −3.90210352360151610014662467825, −3.05433950007339239506292726158, −1.492017670192710040885675316495, −0.249373904294695175643148538083, 1.409911476101312928987102944057, 2.65902496314894249714645840651, 2.94685117853888258927240342978, 4.75110721400653591416193623535, 5.62882688510424310551559974310, 6.422072800544554840718699190045, 7.06589656267802219392884396205, 7.78622123158031861524208154892, 9.19067567871759056706042341474, 9.76449111105989970091298751854, 10.80974138395280999128490404705, 11.69218480885808761958924892427, 12.227829760827753245556525580981, 13.207189939362525405364086691639, 13.90594477661272581904693743506, 14.63388274811965810938267529043, 15.697563631728901730229470236630, 16.490117394358943343242572275734, 17.385391023586737628116510070869, 18.11819647054467349033932253403, 18.701176216412852170464596087984, 19.30415209113112879668754055769, 20.10920669058894495898365972655, 21.45596317545253622912492161299, 22.0528627884815709050567813861

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