L-function 1-760-760.307-r1-0-0

• Label: 1-760-760.307-r1-0-0
• Degree: $1$
• Conductor: $760$
• Sign: $-0.949 + 0.313i$
• Analytic cond.: $81.6733$
• Root an. cond.: $81.6733$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.642 + 0.766i)3^-s + (0.866 + 0.5i)7^-s + (−0.173 + 0.984i)9^-s + (−0.5 − 0.866i)11^-s + (0.642 − 0.766i)13^-s + (−0.984 + 0.173i)17^-s + (0.173 + 0.984i)21^-s + (−0.342 − 0.939i)23^-s + (−0.866 + 0.5i)27^-s + (−0.173 + 0.984i)29^-s + (−0.5 + 0.866i)31^-s + (0.342 − 0.939i)33^-s − i·37^-s + 39^-s + (−0.766 + 0.642i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$760$    =    $2^{3} \cdot 5 \cdot 19$
Sign:	$-0.949 + 0.313i$
Analytic conductor:	$81.6733$
Root analytic conductor:	$81.6733$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{760} (307, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 760,\ (1:\ ),\ -0.949 + 0.313i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.2445691508 + 1.523055175i$
$L(1)$	$\approx$	$1.117162518 + 0.4681846828i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	19	$1$
good	3	$1 + (0.642 + 0.766i)T$
	7	$1 + (0.866 + 0.5i)T$
	11	$1 + (-0.5 - 0.866i)T$
	13	$1 + (0.642 - 0.766i)T$
	17	$1 + (-0.984 + 0.173i)T$
	23	$1 + (-0.342 - 0.939i)T$
	29	$1 + (-0.173 + 0.984i)T$
	31	$1 + (-0.5 + 0.866i)T$
	37	$1 - iT$

Imaginary part of the first few zeros on the critical line

−21.69266341988380691503666934879, −20.68793004306569139928959915533, −20.426887345037161084323388725302, −19.42773311896267974817879811033, −18.61682081645836169422020922775, −17.772058672925977482252442394819, −17.38261067286890057587211717889, −16.037252705274256907346233956223, −15.144694516959840957147945136543, −14.433482663510439513663988987340, −13.50164158411262808441933694348, −13.1280811625554774575600869069, −11.83919169157858481618853519852, −11.31060586310392147955526929589, −10.10338639559708241795827957474, −9.13470281993617862367714313893, −8.30181627459854928488665546848, −7.45361038308710701043360739839, −6.86678259838434411752877023050, −5.68230783535816550614737943952, −4.445302960117796337543262613577, −3.667491192821603342549067419406, −2.15035365565159932632578277056, −1.73634765794197865684927423049, −0.28731868279981875197038002775, 1.44627506461586720620060046611, 2.63337753331083960124332126652, 3.40144643434368221358088772088, 4.59369829341156472475378758247, 5.27581952805658752662599998727, 6.311484553821096590817471022454, 7.75954330983725400750467875540, 8.53245749390288167587251815568, 8.87078950629118737230482413466, 10.2800047890931447800018181107, 10.81736300231293967841590340476, 11.61115664514469167863522476003, 12.935632189778778656095883587920, 13.632802950043949988506274558153, 14.58151464418643215087104567480, 15.17328123648315863555731023226, 15.99242590294608672201899721664, 16.653938415119455683010291677471, 17.95078258899841187464979278589, 18.40983614097568743438117461433, 19.52254497789255605322028571207, 20.3355086876113374184936667860, 20.90751284391703665355269067046, 21.77486436061778322381074862112, 22.20278145678105860868488450173

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