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Properties

Label 1-1045-1045.4-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.0151 - 0.999i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (0.374 − 0.927i)2-s + (−0.961 − 0.275i)3-s + (−0.719 − 0.694i)4-s + (−0.615 + 0.788i)6-s + (−0.913 − 0.406i)7-s + (−0.913 + 0.406i)8-s + (0.848 + 0.529i)9-s + (0.5 + 0.866i)12-s + (−0.0348 + 0.999i)13-s + (−0.719 + 0.694i)14-s + (0.0348 + 0.999i)16-s + (−0.848 + 0.529i)17-s + (0.809 − 0.587i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.990 − 0.139i)24-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s + (−0.961 − 0.275i)3-s + (−0.719 − 0.694i)4-s + (−0.615 + 0.788i)6-s + (−0.913 − 0.406i)7-s + (−0.913 + 0.406i)8-s + (0.848 + 0.529i)9-s + (0.5 + 0.866i)12-s + (−0.0348 + 0.999i)13-s + (−0.719 + 0.694i)14-s + (0.0348 + 0.999i)16-s + (−0.848 + 0.529i)17-s + (0.809 − 0.587i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.990 − 0.139i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0151 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0151 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.0151 - 0.999i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.0151 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5787483559 - 0.5876123364i\)
\(L(\frac12)\) \(\approx\) \(0.5787483559 - 0.5876123364i\)
\(L(1)\) \(\approx\) \(0.6304476196 - 0.4114513263i\)
\(L(1)\) \(\approx\) \(0.6304476196 - 0.4114513263i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.374 - 0.927i)T \)
3 \( 1 + (-0.961 - 0.275i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (-0.0348 + 0.999i)T \)
17 \( 1 + (-0.848 + 0.529i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.241 + 0.970i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.961 + 0.275i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (0.997 - 0.0697i)T \)
53 \( 1 + (0.882 - 0.469i)T \)
59 \( 1 + (-0.997 - 0.0697i)T \)
61 \( 1 + (0.990 + 0.139i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.882 - 0.469i)T \)
73 \( 1 + (-0.559 + 0.829i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.374 - 0.927i)T \)
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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

−22.062639297740253672951395333960, −21.35983934895203063251477061857, −20.33324765141996991282138352336, −19.19250993941193053158013674681, −18.28683216131718077355417686005, −17.64690209870681736965229034559, −16.998393361644352062166323460942, −16.102720864070593134076462096620, −15.53783760270162570449771908066, −15.1309181493126480481098383155, −13.70447600093490824375496638907, −13.122456779634417212347266340404, −12.30227833859180381820291726916, −11.65096936839972481516696565490, −10.45712861719299119394557568121, −9.6210183299032632848797771247, −8.90094090031547383918114090308, −7.689617543313675801686945157890, −6.89876033643579612787174362867, −6.03630105852487732487085281364, −5.55137256077118031044210598296, −4.58762304780852010328247969836, −3.68751979250983943052048151505, −2.67019769055289221416698880791, −0.639843960065329748470682321854, 0.62681340557089485643275964771, 1.783951273462246224308018507490, 2.77383025666942221380779483377, 4.160702494166039706392556347185, 4.4619386243102804438800420805, 5.82627942345600236325235075476, 6.37492115678525844929127005271, 7.26019486974582374049086406627, 8.70034383399679289914230999103, 9.592861532398875085461443039369, 10.367928884038574021223597064776, 11.05935972458363073996209011178, 11.775743970813339735835715823996, 12.687803138315436569141732857895, 13.09200293629506157076314121399, 13.95850933694102476348522195613, 14.92820277198057694263918626148, 16.060314850168677687315033364309, 16.67394270138998706674193410146, 17.60421339447413162553085489153, 18.37705473658716738702163442102, 19.113010458265643656767288774693, 19.68632459528327304743501018813, 20.59615830025442090733432793828, 21.584969263831470272583356069156

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