Properties

Label 1-195-195.173-r0-0-0
Degree $1$
Conductor $195$
Sign $-0.514 - 0.857i$
Analytic cond. $0.905576$
Root an. cond. $0.905576$
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  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)11-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.866 + 0.5i)22-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.5i)28-s + (−0.5 − 0.866i)29-s − 31-s + (0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)11-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.866 + 0.5i)22-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.5i)28-s + (−0.5 − 0.866i)29-s − 31-s + (0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.514 - 0.857i$
Analytic conductor: \(0.905576\)
Root analytic conductor: \(0.905576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (0:\ ),\ -0.514 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1323416844 - 0.2338309119i\)
\(L(\frac12)\) \(\approx\) \(0.1323416844 - 0.2338309119i\)
\(L(1)\) \(\approx\) \(0.4950080514 + 0.02789234401i\)
\(L(1)\) \(\approx\) \(0.4950080514 + 0.02789234401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−27.471839723214024596111329696238, −26.14669319514211693772822488761, −25.8606732362523333987629046489, −24.77861113651900600736639122745, −23.55990365298382697914826608765, −22.19729261651139091624803856884, −21.67731986929370620845449809383, −20.260742568109915468499201003119, −19.78978253174724235609568611606, −18.597321884871178871885611489131, −17.94217744519209753352796633095, −16.82088350800115125048710942994, −15.851612309154096323821574619172, −15.00152392943449888774035255238, −13.099699237042114364750958010068, −12.61724094753276970996342421400, −11.37261054208235866853865486430, −10.31938784791332340540003450379, −9.41149042416254533559496866297, −8.49053513260218190703221745410, −7.21880512951733363761281251621, −6.22759302995466086900592038286, −4.42987198654938389640293333837, −2.98278203540044493867625699944, −1.94275930450997271803684832681, 0.25347666483168864581514220129, 2.14707559297287447371045054684, 3.73479409657764535851107706286, 5.5043716862457026191766996427, 6.45139548585015341940340347578, 7.523107376841975966615094725217, 8.579459742303004074437783621244, 9.67468626930601423433179085613, 10.53744523630254507829582194236, 11.56932086550333371681003961559, 13.12700062726711515371911832925, 14.050488778466177403937778352200, 15.3384358754392334115684861731, 16.219542980265633764377301555724, 16.86348460866313604327877955419, 18.07528825600984806685571406876, 18.911845245428169912426016794943, 19.78415379611901379107365664237, 20.65799484452322663538762481479, 22.01902441379520569670357856749, 23.179464608266459782977535239130, 23.94996732410818193050611662084, 24.96955433531341411641622265363, 25.88088162885975573994117509338, 26.64119019765598805924426882732

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