L-function 1-1045-1045.784-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

$L(\frac{1}{2})$	$\approx$	$0.5787483559 + 0.5876123364i$
$L(1)$	$\approx$	$0.6304476196 + 0.4114513263i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	5	$1$
	11	$1$
	19	$1$
good	2	$1 + (0.374 + 0.927i)T$
	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	$1 + (0.809 + 0.587i)T$

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