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Properties

Label 1-195-195.128-r1-0-0
Degree $1$
Conductor $195$
Sign $0.00863 + 0.999i$
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 + (−0.866 − 0.5i)11-s + 14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)19-s + (−0.866 + 0.5i)22-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)28-s + (−0.5 + 0.866i)29-s + i·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 + (−0.866 − 0.5i)11-s + 14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)19-s + (−0.866 + 0.5i)22-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)28-s + (−0.5 + 0.866i)29-s + i·31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00863 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00863 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.00863 + 0.999i$
Analytic conductor: \(20.9556\)
Root analytic conductor: \(20.9556\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (1:\ ),\ 0.00863 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3294295296 + 0.3322875593i\)
\(L(\frac12)\) \(\approx\) \(0.3294295296 + 0.3322875593i\)
\(L(1)\) \(\approx\) \(0.9027081645 - 0.3215993266i\)
\(L(1)\) \(\approx\) \(0.9027081645 - 0.3215993266i\)

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 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + T \)
53 \( 1 - iT \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (0.866 + 0.5i)T \)
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

−26.31896120252699071521569114034, −25.65901426479690486422532311122, −24.41430488296137161131087463054, −23.75604000091998849501766919024, −22.976261513960077628018674037103, −21.95281077714087960531695320763, −20.90928028363732221465955738178, −20.10772255331826664950992870488, −18.52194003662717627627822418616, −17.58781838116823755772848753644, −16.8814961708247184509830820793, −15.662893962486128315274266614001, −14.981969095201273885510277294939, −13.705544052090871191302655951796, −13.20828084842534165038742228959, −11.87258945183098307438741790977, −10.67605459065479512753507463631, −9.33853251776759295560426381716, −8.00824096331011527353318737858, −7.32339854782730712802154019681, −6.125224240516520728675921410956, −4.81057828818188188754655861131, −4.04865969707339846809660906754, −2.382806578029663504123908343931, −0.12691157149978926865781005243, 1.76396995403028221998956062453, 2.76795892921327012166143255797, 4.19147573915866021593422781122, 5.3268802294272041899551224283, 6.26410014521358779163335078163, 8.18518235568484056996852708764, 9.046702063866169756720520073104, 10.44774507812201059034678155107, 11.1495279665646958689384133381, 12.32438032652137297280765556613, 13.04467622719210382871060460415, 14.271402660239271668454489998311, 15.08217575139324043539494863068, 16.1481940114618084453802098449, 17.81420094682989775612474881371, 18.46001991658395976527863017253, 19.42932994301707826913574326251, 20.46235270774269883254528702568, 21.47250454124569419122906961636, 21.894934796366689862124223798653, 23.17116429424203191717398334414, 24.00695884592199487951435722152, 24.83683186925003735285019524203, 26.22007153017071291257324066770, 27.24728081599262011261521938977

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