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Properties

Label 1-14e2-196.155-r1-0-0
Degree $1$
Conductor $196$
Sign $-0.838 - 0.545i$
Analytic cond. $21.0631$
Root an. cond. $21.0631$
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  + (−0.623 + 0.781i)3-s + (0.623 − 0.781i)5-s + (−0.222 − 0.974i)9-s + (0.222 − 0.974i)11-s + (−0.222 + 0.974i)13-s + (0.222 + 0.974i)15-s + (−0.900 + 0.433i)17-s − 19-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (−0.900 + 0.433i)29-s − 31-s + (0.623 + 0.781i)33-s + (−0.900 + 0.433i)37-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)3-s + (0.623 − 0.781i)5-s + (−0.222 − 0.974i)9-s + (0.222 − 0.974i)11-s + (−0.222 + 0.974i)13-s + (0.222 + 0.974i)15-s + (−0.900 + 0.433i)17-s − 19-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (−0.900 + 0.433i)29-s − 31-s + (0.623 + 0.781i)33-s + (−0.900 + 0.433i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.838 - 0.545i$
Analytic conductor: \(21.0631\)
Root analytic conductor: \(21.0631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (1:\ ),\ -0.838 - 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08615545021 - 0.2902863003i\)
\(L(\frac12)\) \(\approx\) \(0.08615545021 - 0.2902863003i\)
\(L(1)\) \(\approx\) \(0.7446074823 + 0.008627649295i\)
\(L(1)\) \(\approx\) \(0.7446074823 + 0.008627649295i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 - T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + T \)
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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

−27.24179961689412496446524455070, −25.969420106951338367871157848828, −25.156736424400485687665949253091, −24.4483450713982583962892693904, −23.01544070524948069010857869390, −22.680534777124779037887167578176, −21.687374735881685739064196601357, −20.36989657464803266669899627401, −19.30340619225902683499548883987, −18.289674677271588721827326693747, −17.63163511023616875131252333887, −16.90139720460250478225029833817, −15.34049893940278784695034085778, −14.460205207082313881911941218731, −13.1933090194953815520930131217, −12.599005446282757681358818843411, −11.19936233350697582465210639872, −10.51597490925863027738783691461, −9.24218942529652255559552031382, −7.65417278193615440359257070244, −6.830611552971933587649117787552, −5.902889857689886301063714578301, −4.68711642051828558005975465262, −2.759955565962663017300699689201, −1.71936303951779836802085159291, 0.10658467733366176816935816744, 1.77141518808533125474735457593, 3.67566207999706292144552375675, 4.76365918107562221486101726099, 5.75303301148031185286697846444, 6.74649391077346618856647328391, 8.77200246845161722296501162323, 9.18065298545649684587111753943, 10.53844653927700894974911209324, 11.37093904562023735894910100920, 12.53569004555150240645443162111, 13.59115066163545389432103590254, 14.76017367863923306330514803786, 15.89264030773942205608929489742, 16.88421801393743571108294533590, 17.206903933325379429353300489454, 18.60564756747248296940601114781, 19.806667226589133281199898208516, 20.96350409311295913728208673550, 21.57601756498577023958205798638, 22.28427071235816587564803261148, 23.720461802995032004207604892, 24.21188774321766713853417613885, 25.52008494452145033378987109422, 26.47066695760310414772360423539

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