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Properties

Label 1-195-195.188-r1-0-0
Degree $1$
Conductor $195$
Sign $-0.898 - 0.439i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.898 - 0.439i$
Analytic conductor: \(20.9556\)
Root analytic conductor: \(20.9556\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (1:\ ),\ -0.898 - 0.439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1066131370 - 0.4605064177i\)
\(L(\frac12)\) \(\approx\) \(0.1066131370 - 0.4605064177i\)
\(L(1)\) \(\approx\) \(0.6019082194 - 0.2064299943i\)
\(L(1)\) \(\approx\) \(0.6019082194 - 0.2064299943i\)

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

−27.07322777067237010255817186260, −26.10212674159062010265361793568, −25.505349419901520968492563755627, −24.371509517626330557338830719504, −23.34555819054868666171025947753, −22.977060887723737379044652071777, −21.47948250900761870205716068138, −20.32528402649493127875446358377, −19.2202551145598024373299052736, −18.53993040881559955074959161023, −17.33154381880449134713921606411, −16.53069572225668607014630378477, −15.78687518286402182752673106212, −14.58033297424787444220732989782, −13.67981180859799406354166420710, −12.727439645053157882101191648479, −10.90992417365624107688216681857, −10.195560501826546963100859049790, −9.069456604425574876323589565092, −7.8883226171350839379120710150, −7.05567507189745259967341859861, −5.89080204835177424858020282230, −4.769229588327828528698505051563, −3.26203051046407441053657308811, −1.18507463558436345123465135606, 0.21419669371429536654098423177, 2.04956746722813348498384457348, 2.99136925121448590911187363063, 4.413574110858078632755511934092, 5.78795929635437787489868792369, 7.3751983809515824626168788429, 8.47435847233389653820947252957, 9.45779336504256536164470310616, 10.39138688318511796613049379099, 11.4498251191983246773452864274, 12.65126144156897394843029123626, 13.04298591026452458943876604065, 14.68417684022065006390390928196, 15.78605960346109154563698752082, 16.91319765744367162075687621641, 17.894788007023185826202630004586, 18.89901361938555277739815339957, 19.39709075665108812729196505871, 20.82149838141900237590081297304, 21.28520700547836349707546244129, 22.451356171897544684930717181053, 23.20222269651093996882747753090, 24.661980888232541983986168172956, 25.84080158118893784269532121086, 26.20626337427009508989569252981

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