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Properties

Label 1-14e2-196.83-r0-0-0
Degree $1$
Conductor $196$
Sign $0.999 + 0.0320i$
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  + (−0.222 − 0.974i)3-s + (0.222 + 0.974i)5-s + (−0.900 + 0.433i)9-s + (0.900 + 0.433i)11-s + (0.900 + 0.433i)13-s + (0.900 − 0.433i)15-s + (−0.623 + 0.781i)17-s + 19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)27-s + (0.623 − 0.781i)29-s + 31-s + (0.222 − 0.974i)33-s + (0.623 − 0.781i)37-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)3-s + (0.222 + 0.974i)5-s + (−0.900 + 0.433i)9-s + (0.900 + 0.433i)11-s + (0.900 + 0.433i)13-s + (0.900 − 0.433i)15-s + (−0.623 + 0.781i)17-s + 19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (0.623 + 0.781i)27-s + (0.623 − 0.781i)29-s + 31-s + (0.222 − 0.974i)33-s + (0.623 − 0.781i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0320i$
Analytic conductor: \(0.910220\)
Root analytic conductor: \(0.910220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (0:\ ),\ 0.999 + 0.0320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.151897437 + 0.01846480850i\)
\(L(\frac12)\) \(\approx\) \(1.151897437 + 0.01846480850i\)
\(L(1)\) \(\approx\) \(1.062634850 - 0.05541642426i\)
\(L(1)\) \(\approx\) \(1.062634850 - 0.05541642426i\)

Euler product

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

−27.26964825353521877944875669881, −26.073990994577386374742439957434, −25.0993537950885092883383910491, −24.217987394761054955988534397181, −23.03072086236293749251639914970, −22.15212581166262594571839691605, −21.256150482900790467326650361393, −20.370013127315519673105491676407, −19.731146545396403817424214900912, −18.03365034603580006732043785061, −17.235565458560953398298595418012, −16.11834229882305545113223741297, −15.76796645956051960228017340723, −14.252296459688696470079957201775, −13.41459477988969607638311231116, −11.96910474527471199864878348436, −11.244504677926639986965036881132, −9.87486403842429507613230631806, −9.1135999311749337704719721570, −8.20026750577045448470758948754, −6.346273522032846914006342240417, −5.34609338209689917714212194665, −4.33469609757437709958135281069, −3.185561799259427023043765808421, −1.11554685146817476945626777125, 1.471457234611886892707784819255, 2.66508346484695152673565359372, 4.140097633341566146885486804188, 6.03196418963975376012400833119, 6.54847920683877477377922122717, 7.63795664165295886326806790242, 8.84861583917753398248190931264, 10.24415665976200539168414388544, 11.34900442244654826977308012701, 12.09770448590967394091877779597, 13.45423192861618274324950806123, 14.105625345481893529589798233230, 15.15488985035436775231614268876, 16.539513317996533044598022791761, 17.69083518061208643041314538143, 18.22519321083824909771899491273, 19.239532603777564155914005900098, 20.02012394918405583491628922264, 21.45383476388289002405703047894, 22.533611715213061116308911448219, 23.043906612423406807012166556919, 24.225814374018261993405159903468, 25.05049048397181649458983402931, 25.97339809716252495748469840088, 26.772823970506619461954341494882

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