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Properties

Label 1-14e2-196.111-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: $\chi_{196} (111, \cdot )$
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

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

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