L-function 1-1312-1312.141-r0-0-0

• Label: 1-1312-1312.141-r0-0-0
• Degree: $1$
• Conductor: $1312$
• Sign: $0.772 + 0.634i$
• Analytic cond.: $6.09290$
• Root an. cond.: $6.09290$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.707 − 0.707i)3^-s + (−0.453 − 0.891i)5^-s + (−0.587 + 0.809i)7^-s + i·9^-s + (−0.891 − 0.453i)11^-s + (−0.987 − 0.156i)13^-s + (−0.309 + 0.951i)15^-s + (−0.309 − 0.951i)17^-s + (−0.156 − 0.987i)19^-s + (0.987 − 0.156i)21^-s + (−0.587 − 0.809i)23^-s + (−0.587 + 0.809i)25^-s + (0.707 − 0.707i)27^-s + (−0.891 + 0.453i)29^-s + (0.309 + 0.951i)31^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1312$    =    $2^{5} \cdot 41$
Sign:	$0.772 + 0.634i$
Analytic conductor:	$6.09290$
Root analytic conductor:	$6.09290$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1312} (141, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1312,\ (0:\ ),\ 0.772 + 0.634i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.2146470902 + 0.07685166479i$
$L(1)$	$\approx$	$0.4989837572 - 0.1819965802i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	41	$1$
good	3	$1 + (-0.707 - 0.707i)T$
	5	$1 + (-0.453 - 0.891i)T$
	7	$1 + (-0.587 + 0.809i)T$
	11	$1 + (-0.891 - 0.453i)T$
	13	$1 + (-0.987 - 0.156i)T$
	17	$1 + (-0.309 - 0.951i)T$
	19	$1 + (-0.156 - 0.987i)T$
	23	$1 + (-0.587 - 0.809i)T$
	29	$1 + (-0.891 + 0.453i)T$

Imaginary part of the first few zeros on the critical line

−20.90811165585777995691699236766, −20.17856793225610041230269282292, −19.31000925986905995609466049798, −18.6423112432577997143128229275, −17.63571727350324296272233165291, −17.09784261530932114749603340141, −16.29985995764698604262070095642, −15.46403236960190285519908194695, −14.99474679920181729326448522420, −14.13052772833812627711317160459, −13.04791293613089296252270671417, −12.28056448906004834692684650433, −11.41740559660518526385796275992, −10.66857869290661369850282314122, −10.0119338744743068827429081288, −9.6308767939628147263265168324, −8.01616071848893038668715049611, −7.41202974368085955390132445955, −6.47554920066436968472345677358, −5.80355126164614170800574421714, −4.62403721311329004579927015662, −3.923997277182725325643071614142, −3.191366486513877945194135045162, −1.964751628062722108095425531764, −0.14893843458435880532725223645, 0.68015206138858719359590062456, 2.183685851036947863258735499034, 2.83139449761118354133408659031, 4.34532289777646478921686563993, 5.301504280352666543282740756432, 5.58851930623385207989802108696, 6.899759207133844348353228973004, 7.45270321136233246889044247492, 8.51939060071453070367798902221, 9.08684641952328856880796523689, 10.235675800381705761148341021687, 11.105012328224992357183040073944, 12.076380564755421761681147510652, 12.370172310005923092553605141511, 13.157009830139357465013163192, 13.828277739418395364397777199676, 15.16930765274702730142484963895, 15.92063946925295276511552665058, 16.3947581482232998318767939626, 17.230439403747362227984352343364, 18.1024462342046222961617404504, 18.683003363958245143021767701330, 19.52359828648502667572099053725, 20.0409980078837450260866123716, 21.114371113269004655646951731846

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