L-function 1-273-273.44-r0-0-0

• Label: 1-273-273.44-r0-0-0
• Degree: $1$
• Conductor: $273$
• Sign: $0.349 - 0.936i$
• Analytic cond.: $1.26780$
• Root an. cond.: $1.26780$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.866 − 0.5i)2^-s + (0.5 − 0.866i)4^-s + (0.866 − 0.5i)5^-s − i·8^-s + (0.5 − 0.866i)10^-s + (0.866 + 0.5i)11^-s + (−0.5 − 0.866i)16^-s + (−0.5 + 0.866i)17^-s + (−0.866 + 0.5i)19^-s − i·20^-s + 22^-s + (−0.5 − 0.866i)23^-s + (0.5 − 0.866i)25^-s − 29^-s + (0.866 + 0.5i)31^-s + (−0.866 − 0.5i)32^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$273$    =    $3 \cdot 7 \cdot 13$
Sign:	$0.349 - 0.936i$
Analytic conductor:	$1.26780$
Root analytic conductor:	$1.26780$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{273} (44, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 273,\ (0:\ ),\ 0.349 - 0.936i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.898030195 - 1.317295258i$
$L(1)$	$\approx$	$1.718921942 - 0.7569514352i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−25.7094905529654810775386008991, −24.89507906088240984610549672591, −24.18997364710749574652796349579, −23.062101943205842737729939233908, −22.18296565159319623604921052358, −21.67893603859083278274066073828, −20.717874666194723125453596802916, −19.64341744088417422567679471163, −18.36422673125357417717716596564, −17.34790615088346621855655675605, −16.702714564985387829120435804089, −15.47800651309281819808603188359, −14.65471046304922140208226252619, −13.72079018379754602788880681554, −13.199401829271052854336583554261, −11.79073126046544048830562922507, −11.06117382358985424906276855102, −9.65858729171633752112231778210, −8.58494244674683300563381480293, −7.19666914184150368575570591791, −6.39415248053309884703327714406, −5.53707206997284632578239447032, −4.28489618161643468522049171341, −3.088409274252249262447154656200, −1.96223697104925034890612309862, 1.44815680842568669196518181619, 2.32555232006560521208377276346, 3.91581484910928015210052074736, 4.761715378040968313197858678066, 6.023433925372479159635151544283, 6.64651441161728679079050484816, 8.4415575152061840633210008116, 9.61969747522446328116479722078, 10.38409416953632651163662814742, 11.55800998283664896910639922310, 12.632321456603661489722909384993, 13.15949590118382373158030216137, 14.36596392464286858292501617751, 14.90807209651236337507769459089, 16.28868046864084652870402278093, 17.165655276987720880873107489547, 18.26512336659239815369669729514, 19.47322247435149370802082633636, 20.21445271835393410526363949035, 21.11436864056333468604979821610, 21.84512256893405610963135685573, 22.64914351897255866638569922265, 23.66384238374497181642996398101, 24.63945901852297801364352180874, 25.15929577659941670158077186026

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