L-function 1-76-76.31-r0-0-0

• Label: 1-76-76.31-r0-0-0
• Degree: $1$
• Conductor: $76$
• Sign: $-0.813 + 0.582i$
• Analytic cond.: $0.352942$
• Root an. cond.: $0.352942$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.5 + 0.866i)3^-s + (−0.5 + 0.866i)5^-s − 7^-s + (−0.5 − 0.866i)9^-s − 11^-s + (0.5 + 0.866i)13^-s + (−0.5 − 0.866i)15^-s + (−0.5 + 0.866i)17^-s + (0.5 − 0.866i)21^-s + (0.5 + 0.866i)23^-s + (−0.5 − 0.866i)25^-s + 27^-s + (0.5 + 0.866i)29^-s + 31^-s + (0.5 − 0.866i)33^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$76$    =    $2^{2} \cdot 19$
Sign:	$-0.813 + 0.582i$
Analytic conductor:	$0.352942$
Root analytic conductor:	$0.352942$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{76} (31, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 76,\ (0:\ ),\ -0.813 + 0.582i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.1513100478 + 0.4711510755i$
$L(1)$	$\approx$	$0.5444585745 + 0.3554927930i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−30.96453725105437204053826179350, −29.65968366310593268952569635033, −28.731284905853535931558904026039, −28.09368364796693274089527945158, −26.593525980923935541086053795573, −25.212662284167302064462573807574, −24.44243922779136480881675444591, −23.16953424639576405869516568038, −22.7064726909371224521727937056, −20.87123947515133948945018891343, −19.78498907847391683258141524432, −18.788972818306042386113536472942, −17.65724732039485731661817504769, −16.37203995166865835259410108279, −15.61374586037738500121590160878, −13.438080816455064444966024837614, −12.87438834997580717123564238752, −11.783820688072526194670827910777, −10.3684854346823042521354082626, −8.66032633146557685740066672257, −7.533294131612715784735581358914, −6.14007255244981134512812111322, −4.84961988064021471629710131160, −2.80722110358583131167560499354, −0.58643777676603813148578083397, 3.00082467881282824216785862619, 4.15290600150176723047257843734, 5.87757034690246130943089346732, 7.019664629035192349384364112658, 8.8581221353773641045339907238, 10.24687219894998237649489227321, 10.99746765501690092284644137077, 12.33300278153568286834102221757, 13.89513595701432375219917248887, 15.4487651538860355056711621140, 15.85937534171682304035638596682, 17.254153142848586422501757409837, 18.599214968113827970794717877471, 19.606763176519492967335996780085, 21.09570815109919306970357185341, 22.03149539372091202591994665187, 23.02462314323435186793988287226, 23.74014861695812710581901882900, 25.956065664355465041186329675565, 26.20576025758494651494936786084, 27.42585564007553546541564237415, 28.6154775272682322051572108980, 29.33340741935054888222392611183, 30.88907762195413766939418563356, 31.7612094107870519046522065940

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