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Properties

Label 1-1045-1045.784-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: $\chi_{1045} (784, \cdot )$
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

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

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