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Properties

Label 1-1012-1012.135-r0-0-0
Degree $1$
Conductor $1012$
Sign $0.291 + 0.956i$
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  + (−0.974 − 0.226i)3-s + (−0.897 + 0.441i)5-s + (−0.870 + 0.491i)7-s + (0.897 + 0.441i)9-s + (0.993 − 0.113i)13-s + (0.974 − 0.226i)15-s + (0.254 − 0.967i)17-s + (−0.362 + 0.931i)19-s + (0.959 − 0.281i)21-s + (0.610 − 0.791i)25-s + (−0.774 − 0.633i)27-s + (−0.362 − 0.931i)29-s + (−0.516 − 0.856i)31-s + (0.564 − 0.825i)35-s + (−0.696 + 0.717i)37-s + ⋯
L(s)  = 1  + (−0.974 − 0.226i)3-s + (−0.897 + 0.441i)5-s + (−0.870 + 0.491i)7-s + (0.897 + 0.441i)9-s + (0.993 − 0.113i)13-s + (0.974 − 0.226i)15-s + (0.254 − 0.967i)17-s + (−0.362 + 0.931i)19-s + (0.959 − 0.281i)21-s + (0.610 − 0.791i)25-s + (−0.774 − 0.633i)27-s + (−0.362 − 0.931i)29-s + (−0.516 − 0.856i)31-s + (0.564 − 0.825i)35-s + (−0.696 + 0.717i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.291 + 0.956i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ 0.291 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4703165260 + 0.3483939342i\)
\(L(\frac12)\) \(\approx\) \(0.4703165260 + 0.3483939342i\)
\(L(1)\) \(\approx\) \(0.6080425374 + 0.07225723064i\)
\(L(1)\) \(\approx\) \(0.6080425374 + 0.07225723064i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.974 - 0.226i)T \)
5 \( 1 + (-0.897 + 0.441i)T \)
7 \( 1 + (-0.870 + 0.491i)T \)
13 \( 1 + (0.993 - 0.113i)T \)
17 \( 1 + (0.254 - 0.967i)T \)
19 \( 1 + (-0.362 + 0.931i)T \)
29 \( 1 + (-0.362 - 0.931i)T \)
31 \( 1 + (-0.516 - 0.856i)T \)
37 \( 1 + (-0.696 + 0.717i)T \)
41 \( 1 + (0.696 + 0.717i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.736 + 0.676i)T \)
59 \( 1 + (-0.198 + 0.980i)T \)
61 \( 1 + (0.921 + 0.389i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.564 + 0.825i)T \)
73 \( 1 + (-0.998 - 0.0570i)T \)
79 \( 1 + (0.993 - 0.113i)T \)
83 \( 1 + (-0.985 + 0.170i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.985 + 0.170i)T \)
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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

−21.531508091247678162712052926576, −20.74744803755151885807234757187, −19.79341123519110141643840964611, −19.22244900534346106399176045086, −18.32304025458562916410596216245, −17.37884536593941213252116321596, −16.66894645002459177244777634063, −15.97743397377198767690021445089, −15.58807235672341234414450362234, −14.47332438715402126847792229126, −13.114143549205062964240715532408, −12.76095138818257409804587362308, −11.902204105630456902103691267536, −10.88068856481369451368965596544, −10.60355448680802288441974856785, −9.33467200889050264738092927407, −8.6166309907812776449773272811, −7.396811539162328445820207764348, −6.71271182215563509708232675306, −5.84269214344172872773020041228, −4.8524709404858555334830583569, −3.91596078336015616557851794963, −3.39836960036581233536290795720, −1.50204397555561990980042992936, −0.414619379167224130182573603479, 0.850916976893818897567649615125, 2.33793823684287283320140939952, 3.50424827669038989513133950318, 4.238031175159574410611765177285, 5.52689443529951864687518474524, 6.15858672692952957438824315904, 7.00364996430366529405770037704, 7.77892505489898814891350966363, 8.810804235646319984149615920021, 9.95262228813475317135065137158, 10.6292954462784553419813183471, 11.65148306502651581981178342986, 11.95535383106858971324569633220, 12.94240775042575100868800977357, 13.65711671872532875781798810248, 14.960942418247404352156523095363, 15.64837632458998951805109686846, 16.28722577843613789515873068208, 16.89389315757944026496285965570, 18.15244580496240608668287933821, 18.677617873116570931415116588546, 19.08059689919280661295189935468, 20.19552582727408876634835481989, 21.072141350151738340191274301219, 22.09161188901753873097547535675

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