↑

Properties

Label 1-76-76.27-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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

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 + ⋯
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

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

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: $\chi_{76} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 76,\ (0:\ ),\ -0.813 - 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1513100478 - 0.4711510755i\)
\(L(\frac12)\) \(\approx\) \(0.1513100478 - 0.4711510755i\)
\(L(1)\) \(\approx\) \(0.5444585745 - 0.3554927930i\)
\(L(1)\) \(\approx\) \(0.5444585745 - 0.3554927930i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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