Properties

Label 1-195-195.59-r0-0-0
Degree $1$
Conductor $195$
Sign $0.0257 - 0.999i$
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 + (0.866 − 0.5i)11-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.866 + 0.5i)28-s + (0.5 + 0.866i)29-s + i·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 + (0.866 − 0.5i)11-s − 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.866 + 0.5i)28-s + (0.5 + 0.866i)29-s + i·31-s + (−0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.0257 - 0.999i$
Analytic conductor: \(0.905576\)
Root analytic conductor: \(0.905576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (0:\ ),\ 0.0257 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.256294852 - 1.224305641i\)
\(L(\frac12)\) \(\approx\) \(1.256294852 - 1.224305641i\)
\(L(1)\) \(\approx\) \(1.387362643 - 0.7269846029i\)
\(L(1)\) \(\approx\) \(1.387362643 - 0.7269846029i\)

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

−27.08115155271432035002731438025, −25.84668401816672636725447722105, −25.34973004355792483634545451523, −24.43428758723975984465861559668, −23.2958768720573898943988684744, −22.57743614755793611812141568159, −21.78317739301474942199745373190, −20.79767289120938746602614172122, −19.68469705176277737621698312016, −18.67761254818519752392194536571, −17.132190597451522861989771629779, −16.631027915955626982511100982509, −15.30056533214301687274675845484, −14.7847105956551061669749813913, −13.51919665700045208533840390361, −12.54189334343548385295662650625, −11.91128709468472217322783130692, −10.42705561500374013601405894262, −9.09051307745146738800050332253, −7.952850116661571758861855924499, −6.57207516499453041410418140588, −6.0117381439960904512522067344, −4.51829802446415876238412538098, −3.49662202165266072952629168802, −2.16491337024767705028967633920, 1.140297813995722864991338368178, 2.891150760128994595717554720293, 3.77423682684489977941475367980, 5.036690305805606176443692809993, 6.3315020483291006831805009499, 7.11915033153914875950616796938, 9.009588228836336268792860218299, 10.01639259573617641371105245948, 11.07544141087182198794564096469, 12.05219272065268599823574556444, 13.10538209987938446757880065018, 13.88292155115143510163364506776, 14.86933347391434887551644214430, 16.04004736093054750705227445850, 16.84572830617003336120487108533, 18.39792954209278934934922117495, 19.57672035150112445841341605086, 19.896096936248123503359965778608, 21.28989975554243620562044331952, 21.952734428075047565692141914649, 23.04924326831653607222509444679, 23.54591286966217343553114863744, 24.85813290874507855998382471789, 25.52727822962729418628525424964, 26.94022939603502725875102428691

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