L-function 1-13e2-169.139-r0-0-0

• Label: 1-13e2-169.139-r0-0-0
• Degree: $1$
• Conductor: $169$
• Sign: $-0.981 - 0.190i$
• Analytic cond.: $0.784832$
• Root an. cond.: $0.784832$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.632 + 0.774i)2^-s + (−0.996 − 0.0804i)3^-s + (−0.200 − 0.979i)4^-s + (0.120 + 0.992i)5^-s + (0.692 − 0.721i)6^-s + (−0.845 + 0.534i)7^-s + (0.885 + 0.464i)8^-s + (0.987 + 0.160i)9^-s + (−0.845 − 0.534i)10^-s + (0.987 − 0.160i)11^-s + (0.120 + 0.992i)12^-s + (0.120 − 0.992i)14^-s + (−0.0402 − 0.999i)15^-s + (−0.919 + 0.391i)16^-s + (−0.845 + 0.534i)17^-s + (−0.748 + 0.663i)18^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$169$    =    $13^{2}$
Sign:	$-0.981 - 0.190i$
Analytic conductor:	$0.784832$
Root analytic conductor:	$0.784832$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{169} (139, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 169,\ (0:\ ),\ -0.981 - 0.190i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.02840863958 + 0.2948754027i$
$L(1)$	$\approx$	$0.3678951017 + 0.2769907293i$

Euler product

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

	$p$	$F_p(T)$
bad	13	$1$
good	2	$1 + (-0.632 + 0.774i)T$
	3	$1 + (-0.996 - 0.0804i)T$
	5	$1 + (0.120 + 0.992i)T$
	7	$1 + (-0.845 + 0.534i)T$
	11	$1 + (0.987 - 0.160i)T$
	17	$1 + (-0.845 + 0.534i)T$
	19	$1 + (-0.5 + 0.866i)T$
	23	$1 + (-0.5 - 0.866i)T$
	29	$1 + (-0.632 + 0.774i)T$

Imaginary part of the first few zeros on the critical line

−27.474842509296133229592274355533, −26.3918931723366466594261363949, −25.246681890550250414899650736213, −24.15425048675486346599347890227, −22.95895636871480837935317496519, −22.12225643310822401562268957407, −21.266709096614734566111363655037, −20.02191386405381674923945347914, −19.51240762428800708121621486470, −18.04853023087530696272988146995, −17.224303916914819818892176552600, −16.589581603409328044253905592960, −15.72135442056901384011234374564, −13.48841728364054206801037923338, −12.80794086478722363779162605000, −11.82360078172520930103174914086, −10.95721495571070588521966925472, −9.641661293320670738200572858768, −9.15286514377023750618590638819, −7.452555317137602313937558676293, −6.32754383459150072261066045326, −4.69666818499755831247786605373, −3.783654618853439422617234845186, −1.72795687921730348819313867581, −0.338235113139928641635494620079, 1.88070330788894836621418030329, 3.96964960156148239483055064400, 5.69709006041037315800644580641, 6.415066230516519338399260784053, 7.064823157957693981431214280621, 8.690670110192562459626937256659, 9.916171386849875741868969322221, 10.6812426923152857713462373229, 11.80503599780172449689730362844, 13.15444806064403458460301610614, 14.54223268825439836786668851782, 15.38105021268785230499324909443, 16.475079956223772563701512164708, 17.17381397066809977193130959814, 18.39143341822491084941662758240, 18.76498267259147468350228232070, 19.89446874577691914384641849787, 21.96949819817459554837120544812, 22.32748888032837586490643333146, 23.27107449374082553059558555655, 24.322246583509956682075169358571, 25.28773504498254991646944495164, 26.16057697694338674527181835031, 27.1739345905652516535626542479, 27.91682162445675625971908926165

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