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Properties

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 $0$

Origins

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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: $\chi_{76} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 76,\ (0:\ ),\ -0.813 + 0.582i)\)

Particular Values

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

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 \( 1 + (0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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