L-function 1-3040-3040.1189-r0-0-0

• Label: 1-3040-3040.1189-r0-0-0
• Degree: $1$
• Conductor: $3040$
• Sign: $-0.621 + 0.783i$
• Analytic cond.: $14.1177$
• Root an. cond.: $14.1177$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.965 − 0.258i)3^-s − i·7^-s + (0.866 + 0.5i)9^-s + (−0.707 − 0.707i)11^-s + (−0.258 − 0.965i)13^-s + (−0.5 − 0.866i)17^-s + (−0.258 + 0.965i)21^-s + (−0.866 − 0.5i)23^-s + (−0.707 − 0.707i)27^-s + (−0.258 − 0.965i)29^-s + 31^-s + (0.5 + 0.866i)33^-s + (−0.707 − 0.707i)37^-s + i·39^-s + (−0.866 + 0.5i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3040$    =    $2^{5} \cdot 5 \cdot 19$
Sign:	$-0.621 + 0.783i$
Analytic conductor:	$14.1177$
Root analytic conductor:	$14.1177$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3040} (1189, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3040,\ (0:\ ),\ -0.621 + 0.783i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.1812713147 - 0.3753587743i$
$L(1)$	$\approx$	$0.5630467682 - 0.2937834072i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	19	$1$
good	3	$1 + (-0.965 - 0.258i)T$
	7	$1 - iT$
	11	$1 + (-0.707 - 0.707i)T$
	13	$1 + (-0.258 - 0.965i)T$
	17	$1 + (-0.5 - 0.866i)T$
	23	$1 + (-0.866 - 0.5i)T$
	29	$1 + (-0.258 - 0.965i)T$
	31	$1 + T$
	37	$1 + (-0.707 - 0.707i)T$

Imaginary part of the first few zeros on the critical line

−19.33370846494990767488647563389, −18.63290035236114990166948922253, −18.022481871774816738212287463762, −17.47893375338436183833230186689, −16.699768634464777325579686261061, −15.97352646998003084297868883601, −15.371101275320868506215247995938, −14.896127170092081145964930909922, −13.83489876310093731202965492299, −12.92452657173908069992429540097, −12.30856150875615228747533293907, −11.79075741935616802937101375257, −11.08510696720306150969620639216, −10.20782080275872507112374376910, −9.7032099965954732281706190400, −8.84150511467899896470498710161, −8.045572250793326283333714818466, −7.00016978550291005010601163932, −6.428780424418110328380022544, −5.62960815164509549975526886039, −4.93486883820234920937585335095, −4.323404956471981886379756272990, −3.3010598266243829190044246031, −2.120793941514030925602539057275, −1.55551170446913146394166741995, 0.19296232265444952768754726173, 0.76569250895701434167250443741, 1.99325316480063372149294745626, 2.97344375796369345741657811043, 3.97282441316575063378175193675, 4.80860937432631687346766763909, 5.41895571307899717711122765274, 6.2938403907390346803406494911, 6.91702163104017315463789507297, 7.834835306715395626489007840207, 8.180828315930113291799864434780, 9.6187825510614432617598049675, 10.18342171098821789110214092895, 10.87279180657894287851263879536, 11.3482627989838211479077443963, 12.28984735065800199802118970268, 12.91086184349289197852172453562, 13.67741714518352413971277446582, 14.06752582540037598365769181463, 15.440603915360999389607634749320, 15.83893369573086880965220737439, 16.57551300210796462974416566139, 17.30043208278515215262605950597, 17.74802287077381817386467308744, 18.50661231444933741775070107589

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