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Properties

Label 1-199-199.106-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.924 + 0.380i$
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  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.924 + 0.380i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.924 + 0.380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1469894697 + 0.7440243605i\)
\(L(\frac12)\) \(\approx\) \(0.1469894697 + 0.7440243605i\)
\(L(1)\) \(\approx\) \(0.5206458169 + 0.5519637420i\)
\(L(1)\) \(\approx\) \(0.5206458169 + 0.5519637420i\)

Euler product

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

−26.563809876242184258661642313030, −25.522121025378140479342418685434, −24.89755171435428613582621639322, −23.505599210181488530276253698723, −22.443642850822218219356274209232, −21.956168907801099279989345714, −20.54342832249809707786961441235, −19.75825414752544521735408218030, −18.88049688803472737129691748205, −17.8434625124318536382749324533, −17.09940325953980500385105422451, −16.63786786403937798871223734551, −14.37030086216684709932770762952, −13.4342943919137621360384344016, −12.7464315972939747754670856621, −11.775639013171403578337003362064, −10.52056425370332145936188964010, −9.9031924874520858228593800349, −8.5790522712660119583049109332, −7.30770201220124620672071473007, −6.36289674922013229856170536819, −4.910003216207551350231641550351, −3.253803966294038264828088874109, −1.95110735437489263849057012470, −0.767580523362577346650861556158, 1.75711745663112995175485596016, 3.79910482843247240960622917075, 5.29542686981107052361112416909, 5.92072484588087330177718379213, 6.84850609458985735448955314821, 8.66593802128706758286531838891, 9.57028443938122024707482365331, 9.91792574599218157951225474146, 11.405734773084386781598321892812, 12.650653145486628651838574001137, 14.33231473408423528287292740365, 14.66987152096401497195391728968, 16.04622581566455981563379138401, 16.71630069655017549701091520184, 17.42653240146601586858361957614, 18.49541451189595412990555744808, 19.43510707974844860131874151924, 20.91465538532739607764951763118, 21.94433306870098588058897364360, 22.43067351830648368690681211568, 23.63440292537996071474982178044, 24.76412211705129438200228993053, 25.60735553112645430006350713428, 26.1717308981049779125501144614, 27.4887861183011682475795930246

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