Properties

Label 1-199-199.139-r0-0-0
Degree $1$
Conductor $199$
Sign $-0.181 - 0.983i$
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  + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.654 − 0.755i)5-s + (0.415 + 0.909i)6-s + (−0.142 − 0.989i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.959 − 0.281i)10-s + (−0.959 − 0.281i)11-s + (0.841 + 0.540i)12-s + (0.841 − 0.540i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)15-s + (−0.654 − 0.755i)16-s + (−0.959 − 0.281i)17-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.654 − 0.755i)5-s + (0.415 + 0.909i)6-s + (−0.142 − 0.989i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.959 − 0.281i)10-s + (−0.959 − 0.281i)11-s + (0.841 + 0.540i)12-s + (0.841 − 0.540i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)15-s + (−0.654 − 0.755i)16-s + (−0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(199\)
Sign: $-0.181 - 0.983i$
Analytic conductor: \(0.924152\)
Root analytic conductor: \(0.924152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{199} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 199,\ (0:\ ),\ -0.181 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8604934739 - 1.033511577i\)
\(L(\frac12)\) \(\approx\) \(0.8604934739 - 1.033511577i\)
\(L(1)\) \(\approx\) \(1.158271829 - 0.5522629997i\)
\(L(1)\) \(\approx\) \(1.158271829 - 0.5522629997i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad199 \( 1 \)
good2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (-0.142 + 0.989i)T \)
5 \( 1 + (-0.654 - 0.755i)T \)
7 \( 1 + (-0.142 - 0.989i)T \)
11 \( 1 + (-0.959 - 0.281i)T \)
13 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.415 + 0.909i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−26.79889756477339649901428217566, −25.96145916607038329934297375946, −25.149713376269708635802218352288, −24.20080555784520710209365812796, −23.42984210657895950075930536028, −22.700849328256944660719977175447, −21.87499229317977680092137016979, −20.63959280966023898927664759068, −19.43624030011452068243330669827, −18.35903134429715600214902726238, −17.85741109880335798749486922288, −16.21998672001361386266054420339, −15.52775796570553964806726843723, −14.51963616693621544923835143449, −13.512822462887737140084080862243, −12.58065308164148694476629208145, −11.72618525917641894238104231320, −10.901773928300978753429091429795, −8.69575712909801575625919275723, −7.852691714347533962691373618795, −6.77378614812168174903168290365, −6.07619379427180123875100954390, −4.77771603688513026117341146114, −3.13812209780129305043955799469, −2.29176704055225817559279001057, 0.80940113896331266467280642162, 3.04418694250633865827123840933, 3.94546100955613052834692821714, 4.85030527124059324423025481035, 5.7813983818230793647696380427, 7.45458808018124849913739586516, 8.88556188776394154054813243094, 10.097113608365242428152751018550, 10.96605976285769859440998063015, 11.697901463690631566259473185844, 13.14452957657522229542440803177, 13.70284191525234904545955461436, 15.22291975221970444800397869817, 15.836650234472922318904484770673, 16.5773051400879306808553669794, 18.04572286271203761900859227287, 19.60081879198061434757862082057, 20.34093648175244495219776861116, 20.78053508168047827577292972777, 21.8732030970603061026885933297, 22.99640543794552048794963709979, 23.42702842371196033623108090885, 24.41420667408659174001133040228, 25.79835743877390775851887164056, 26.96274831434574466477564342947

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