L-function 1-13e2-169.131-r0-0-0

• Label: 1-13e2-169.131-r0-0-0
• Degree: $1$
• Conductor: $169$
• Sign: $0.683 - 0.729i$
• 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

+ (0.885 − 0.464i)2^-s + (−0.970 + 0.239i)3^-s + (0.568 − 0.822i)4^-s + (−0.354 + 0.935i)5^-s + (−0.748 + 0.663i)6^-s + (0.120 − 0.992i)7^-s + (0.120 − 0.992i)8^-s + (0.885 − 0.464i)9^-s + (0.120 + 0.992i)10^-s + (0.885 + 0.464i)11^-s + (−0.354 + 0.935i)12^-s + (−0.354 − 0.935i)14^-s + (0.120 − 0.992i)15^-s + (−0.354 − 0.935i)16^-s + (0.120 − 0.992i)17^-s + (0.568 − 0.822i)18^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$169$    =    $13^{2}$
Sign:	$0.683 - 0.729i$
Analytic conductor:	$0.784832$
Root analytic conductor:	$0.784832$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{169} (131, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 169,\ (0:\ ),\ 0.683 - 0.729i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.295598060 - 0.5615133831i$
$L(1)$	$\approx$	$1.268189624 - 0.3373617378i$

Euler product

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

	$p$	$F_p(T)$
bad	13	$1$
good	2	$1 + (0.885 - 0.464i)T$
	3	$1 + (-0.970 + 0.239i)T$
	5	$1 + (-0.354 + 0.935i)T$
	7	$1 + (0.120 - 0.992i)T$
	11	$1 + (0.885 + 0.464i)T$
	17	$1 + (0.120 - 0.992i)T$
	19	$1 + T$
	23	$1 + T$
	29	$1 + (0.885 - 0.464i)T$

Imaginary part of the first few zeros on the critical line

−27.88651261074955061704865888566, −26.86263565286334990856863233543, −25.18461138025577310361833083616, −24.62855251944673883983547288805, −23.89601262508483715314489663455, −22.99299133145382509472184287806, −21.92970965338582813703144467142, −21.41990688612430209258264368017, −20.080536978200867882666499338231, −18.85655905915013400593939334584, −17.487916193172931598425067505797, −16.725429154954688628364858878695, −15.910109046775552576485075367804, −14.97563873382243107654683936161, −13.55483579313420893643207395309, −12.471834705957748519709948289914, −11.988628592613521960092079828925, −11.05227199500354124199188513247, −9.096020311083055520288477304120, −8.00454428967281893202703812754, −6.66968574300465132629524808503, −5.62163805301369504840836504991, −4.917635047125664308733281570708, −3.596845655541757622639811907544, −1.60070110083289774435875825399, 1.206662595099503854441920780901, 3.190575916457331903307589581916, 4.20611890135551728995030399679, 5.246548712455300933791396628265, 6.79123755035835885647126819325, 7.10829868914295890136482949034, 9.72392738058294178653893188928, 10.514947090249089956835455494622, 11.466545234169376286686742061250, 12.04965118082563983880994410141, 13.52032984192250304775897807885, 14.42834651011740935809030168115, 15.45066181769036749282507417135, 16.452511428393249870979171665183, 17.65293264448484990097870355618, 18.73122203731236945824236301620, 19.8603968550645965826285770775, 20.78911066674607880172938691370, 21.97896420894474844619770758347, 22.71537289190267517327239171978, 23.17330606765561753117191155195, 24.138918772191026467433390370968, 25.3534472813370016912795997317, 27.0188845374965916161785357972, 27.31350296179807313186460152875

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