Properties

Label 1-14e2-196.139-r0-0-0
Degree $1$
Conductor $196$
Sign $0.545 - 0.838i$
Analytic cond. $0.910220$
Root an. cond. $0.910220$
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

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

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(0.910220\)
Root analytic conductor: \(0.910220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (0:\ ),\ 0.545 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.139204883 - 0.6177499204i\)
\(L(\frac12)\) \(\approx\) \(1.139204883 - 0.6177499204i\)
\(L(1)\) \(\approx\) \(1.140761409 - 0.3240909408i\)
\(L(1)\) \(\approx\) \(1.140761409 - 0.3240909408i\)

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

−27.05619557700063471060283556866, −26.31255630973323483647357772139, −25.279094881183435246845885640266, −24.4382619513088909055187187781, −23.282156549036364342163093086730, −22.40151821439710206116406110653, −21.06160064117971498041700777573, −20.639307710977433287273809046663, −19.61369349014655897460542845188, −18.8242607868757247335426197459, −17.16425082743674343455713219440, −16.4167480362273232817206072077, −15.48817009038667697606784334660, −14.64396694732388099180638802579, −13.55060267730917477801131327513, −12.33440460518324099084237671978, −11.37623883233559850133817792465, −9.986665323954010452431356711128, −9.174611228472744424051805681651, −8.20793778945674048176538786047, −7.13203917488521392125303362816, −5.275202394176214995525873434383, −4.36517050624453500345056252003, −3.39435667596992394604599919052, −1.697498561208820234200921020833, 1.08024334080341433465582389671, 3.004018843256696815075142157995, 3.430128377215506464036241252607, 5.50073853964195736523981085774, 6.75716753756006890042337559161, 7.68024551073316661925038902655, 8.49436588001092689185348676684, 9.85551776899075159135271535382, 11.19822429913628135180759027389, 12.01507895432693730947627738433, 13.26244294623397136005776192617, 14.12521528790590513601469868935, 15.01742711140099039971313841662, 15.99797799018921524507080552655, 17.41892513969925667857326600063, 18.54242614748955634819482596162, 18.99927177488565777007213081043, 20.00410670681670139709290928630, 20.91936006832704005692117404746, 22.300161697620524807590107191520, 23.10904617198153141631858827629, 24.05228399433815909207572351301, 24.96916764145765372810347611962, 25.824711700486934783492852381645, 26.82283943866104194413698373812

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