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Properties

Label 1-195-195.134-r1-0-0
Degree $1$
Conductor $195$
Sign $0.0128 - 0.999i$
Analytic cond. $20.9556$
Root an. cond. $20.9556$
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)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)11-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)28-s + (0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)11-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)28-s + (0.5 + 0.866i)29-s − 31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(20.9556\)
Root analytic conductor: \(20.9556\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (1:\ ),\ 0.0128 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04036695815 - 0.03985259632i\)
\(L(\frac12)\) \(\approx\) \(0.04036695815 - 0.03985259632i\)
\(L(1)\) \(\approx\) \(0.7734993111 + 0.4599048924i\)
\(L(1)\) \(\approx\) \(0.7734993111 + 0.4599048924i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + T \)
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 \( 1 + T \)
79 \( 1 + T \)
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

−27.14351099242078852693584203610, −26.23041867278259689690019835793, −25.01845580894510599260889312287, −23.806289074991558832496463727561, −23.01474038993171844684018127757, −22.39645030180865818857101755815, −21.14396190338881492112056662922, −20.30331921125632426642708166178, −19.689755230744162388815703318848, −18.47825752970611490146948230203, −17.635314728581786437351751275275, −16.237316266939445392295648745965, −15.180357385944424282852557790889, −13.97816844547600938687113976085, −13.279554618710459860862269681602, −12.265360012985461053283061996539, −11.23623877534812642893270220483, −10.07605993014298697161986098281, −9.53393641390114402369254317917, −7.80509673488507874449194691664, −6.54497083420190993477097233400, −5.173321288319805652262028985987, −4.11047744722598572818375235011, −2.98158282189680574977783247169, −1.56099968771323992192225360619, 0.01599015317802703654715240319, 2.56131871147668518363280159347, 3.71397524789348630450427302332, 5.17743418041809450499925381689, 6.01873887717105346198091711098, 7.07274065591206591782673410919, 8.44938752208801562682725370047, 9.067634347979663950543442975022, 10.687337706531032134333893274160, 12.06491145104062307871126041836, 12.916864194716775659839329157224, 13.8518911171774630114493259050, 14.98217375703875362218300565129, 15.83431449570395561589178808742, 16.53646397253016858367430525879, 17.82622539279848824922934023044, 18.59141850197650169480036127370, 19.799342208739800791900469680717, 21.26356800135504599485052193284, 21.92454617415189911077530507215, 22.689775705668213677554681390246, 24.01219742140349823298829853774, 24.38783797566834859471827877901, 25.65862329264384950340240202931, 26.19445502023629725750375969926

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