L-function 1-153-153.128-r1-0-0

• Label: 1-153-153.128-r1-0-0
• Degree: $1$
• Conductor: $153$
• Sign: $0.133 - 0.991i$
• Analytic cond.: $16.4421$
• Root an. cond.: $16.4421$
• 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 + (−0.965 + 0.258i)5^-s + (0.965 + 0.258i)7^-s − i·8^-s + (−0.707 + 0.707i)10^-s + (0.965 + 0.258i)11^-s + (0.5 − 0.866i)13^-s + (0.965 − 0.258i)14^-s + (−0.5 − 0.866i)16^-s − i·19^-s + (−0.258 + 0.965i)20^-s + (0.965 − 0.258i)22^-s + (−0.258 − 0.965i)23^-s + (0.866 − 0.5i)25^-s − i·26^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$153$    =    $3^{2} \cdot 17$
Sign:	$0.133 - 0.991i$
Analytic conductor:	$16.4421$
Root analytic conductor:	$16.4421$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{153} (128, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 153,\ (1:\ ),\ 0.133 - 0.991i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.212266218 - 1.934443494i$
$L(1)$	$\approx$	$1.620936543 - 0.7207802631i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−27.65930762894801043299514850089, −27.09405859057722759998659092174, −25.888527472903000995198820240075, −24.75221821800092080021113774250, −23.92530307798550989904602110704, −23.3617690593912335574606300448, −22.22491038270832260302040843386, −21.17112183500569401665047788891, −20.33054450243347006833477507778, −19.251680267741497819801832496006, −17.78660676670954314860419795006, −16.63284577615338912319497342969, −15.98126738402049672616496530197, −14.630588642769430325310806469635, −14.17240858968253655051935182375, −12.73024935141601208901072241864, −11.65702159103913906331897808541, −11.128433035818366691190593735741, −8.9430365901961284818897905497, −7.94220192611283077665258609079, −7.00076322547200261611861619707, −5.615073648967271889409299597578, −4.28807041628245694974749295453, −3.65665658595157164399827898339, −1.57811761314141607038018843056, 0.941544432252174846541596478642, 2.57378017092832365170150701060, 3.92150926059061505608007007310, 4.81367818506680757380820608441, 6.23892357889258505652053277778, 7.501579288627620838848828563512, 8.80542535521314255906904474936, 10.45228504515983200150616680990, 11.39607860786312450875825484509, 12.03529990268774483222276887132, 13.24965903589167821202132970506, 14.59997900311599543653835682487, 15.0631082179336011853221162088, 16.16661287333566998382996772798, 17.75842006843718802214660967330, 18.83531363406225120079900979787, 19.924187426890430932003845449680, 20.5216718201839889224307415435, 21.79993957670440535113527325331, 22.589641639745033466068804590431, 23.51728123241874873390198374828, 24.36184073459540247515914319509, 25.25509623502611436525607620561, 26.86489281787507494010424200312, 27.79879663863715783887894097643

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