Properties

Label 1-87-87.26-r0-0-0
Degree $1$
Conductor $87$
Sign $-0.721 + 0.692i$
Analytic cond. $0.404026$
Root an. cond. $0.404026$
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  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.974 − 0.222i)8-s + (−0.781 − 0.623i)10-s + (−0.974 + 0.222i)11-s + (0.222 + 0.974i)13-s + (−0.433 + 0.900i)14-s + (−0.222 − 0.974i)16-s i·17-s + (0.781 + 0.623i)19-s + (0.222 − 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (0.623 + 0.781i)7-s + (−0.974 − 0.222i)8-s + (−0.781 − 0.623i)10-s + (−0.974 + 0.222i)11-s + (0.222 + 0.974i)13-s + (−0.433 + 0.900i)14-s + (−0.222 − 0.974i)16-s i·17-s + (0.781 + 0.623i)19-s + (0.222 − 0.974i)20-s + (−0.623 − 0.781i)22-s + (0.900 + 0.433i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.721 + 0.692i$
Analytic conductor: \(0.404026\)
Root analytic conductor: \(0.404026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (0:\ ),\ -0.721 + 0.692i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3542728196 + 0.8807266890i\)
\(L(\frac12)\) \(\approx\) \(0.3542728196 + 0.8807266890i\)
\(L(1)\) \(\approx\) \(0.7553745800 + 0.7030482866i\)
\(L(1)\) \(\approx\) \(0.7553745800 + 0.7030482866i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.433 + 0.900i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.974 + 0.222i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.781 + 0.623i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.433 - 0.900i)T \)
37 \( 1 + (0.974 + 0.222i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.433 - 0.900i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.781 + 0.623i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.433 + 0.900i)T \)
79 \( 1 + (-0.974 - 0.222i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.433 + 0.900i)T \)
97 \( 1 + (-0.781 - 0.623i)T \)
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

−30.5559146149552418690614380661, −29.21143308931650183089577392818, −28.222114545118318319215730850424, −27.2934409928680765237649363120, −26.45163683359203150691191355614, −24.47866581327866012080684575341, −23.60813800656553742635940359292, −22.940808776774456325027437955543, −21.48068570754627831342105370035, −20.457044212323616829372341846323, −19.83479415710897676135723430995, −18.58145858638872266808485952558, −17.40753558669617481653298901576, −15.8045823142082642563770832819, −14.747780353804784180658261718811, −13.36170877342382247441106825292, −12.53003184622763307443503758391, −11.13083534946599178701667202623, −10.50119772771181409105468632491, −8.72469479673940643525755678228, −7.57842496795840031996779403808, −5.42643307801173058124215774741, −4.34122125374951387539342939255, −3.06009331908932082598530103317, −0.982978111281199261678279721400, 2.848667571574123231337718239800, 4.40013847907254126495862282986, 5.565937861276482788966325519870, 7.15414324291779301247650582347, 7.999215966436878882566264230806, 9.284557523354463001135296268, 11.30755405435786989461598581007, 12.19077511250837601108800440245, 13.65729432296885366032889237918, 14.82179835625176955632391453725, 15.59976323226412685677486134995, 16.568641092988247143135442200585, 18.22741400886533025441298701494, 18.678722310701522983815328721061, 20.58863458548226820049009144670, 21.63493043932739930777472592838, 22.79768433191037581555066619095, 23.627980800399616680911214943790, 24.55275137374141341570842157320, 25.69134435347347885529358425003, 26.762282525600077721482367613, 27.50411794465355672853204970652, 28.884428474762510969817586230222, 30.52871386700763659797827243470, 31.31979488357886275912369330146

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