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Properties

Label 1-199-199.175-r0-0-0
Degree $1$
Conductor $199$
Sign $0.961 - 0.274i$
Analytic cond. $0.924152$
Root an. cond. $0.924152$
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  + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.5 + 0.866i)5-s + (0.766 − 0.642i)6-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.766 + 0.642i)10-s + 11-s + (−0.5 − 0.866i)12-s + (0.766 + 0.642i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)15-s + (0.766 + 0.642i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.5 + 0.866i)5-s + (0.766 − 0.642i)6-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.766 + 0.642i)10-s + 11-s + (−0.5 − 0.866i)12-s + (0.766 + 0.642i)13-s + (−0.939 − 0.342i)14-s + (−0.939 + 0.342i)15-s + (0.766 + 0.642i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $0.961 - 0.274i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ 0.961 - 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.438161457 - 0.2011101971i\)
\(L(\frac12)\) \(\approx\) \(1.438161457 - 0.2011101971i\)
\(L(1)\) \(\approx\) \(1.281825395 - 0.2305965121i\)
\(L(1)\) \(\approx\) \(1.281825395 - 0.2305965121i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−26.77074335804570313360746293781, −25.6943701698996934307371812858, −24.83699772255159344107785437316, −24.47368785027773511548321475465, −23.55250762404045230875014097227, −22.44188856419113045739202940809, −21.319828889940260813712337963014, −20.176803203900094970907990955854, −19.2383183902651341721688847525, −18.23212164593841697183002714969, −17.375714776886948357416229651017, −16.13119199168263468078874883392, −15.23858208064577096176702795069, −14.55809262077386397519727791667, −13.22251006035641658403873533965, −12.67858760875350521611524112036, −11.6033328271465348774672404420, −9.316382867349338032897886879113, −8.715136756735640702041696106776, −8.037184505829370667218540000067, −6.77447479611848746971034061160, −5.71434526281222654478909209636, −4.38081969377021132619974542405, −3.18987709224218661213701755320, −1.2406198302051021516739330862, 1.574704485024391576362103856241, 3.174257207372100303199189023349, 3.79281557398078028626881477304, 4.76682302914839037436352025572, 6.73328748316112464979479016975, 8.02071820234363262988873906875, 9.21713460751592710165104420026, 10.10993163979889636987356308942, 11.09345709217750695959130430023, 11.752951193410170743696740213398, 13.6340889983759281840001810480, 13.961656591060321190912661351081, 14.92334244858133389875871065598, 16.1100606527365079913635152721, 17.43385412853344007414679257559, 18.71853783991524935859534097222, 19.42534266358782505019134741074, 20.279367443794342708428594767692, 20.97354386196931080553358761401, 22.122621028832599667716012179607, 22.77065301902154522028398420920, 23.72215368647704735533149202034, 25.186884706672083340222761218236, 26.39710940117566046282937694328, 27.0405280556152225858266981585

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