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Properties

Label 1-14e2-196.179-r1-0-0
Degree $1$
Conductor $196$
Sign $0.996 - 0.0853i$
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  + (−0.826 − 0.563i)3-s + (0.0747 − 0.997i)5-s + (0.365 + 0.930i)9-s + (−0.365 + 0.930i)11-s + (0.623 + 0.781i)13-s + (−0.623 + 0.781i)15-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.222 − 0.974i)27-s + (−0.222 − 0.974i)29-s + (0.5 + 0.866i)31-s + (0.826 − 0.563i)33-s + (0.955 + 0.294i)37-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)3-s + (0.0747 − 0.997i)5-s + (0.365 + 0.930i)9-s + (−0.365 + 0.930i)11-s + (0.623 + 0.781i)13-s + (−0.623 + 0.781i)15-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.222 − 0.974i)27-s + (−0.222 − 0.974i)29-s + (0.5 + 0.866i)31-s + (0.826 − 0.563i)33-s + (0.955 + 0.294i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.996 - 0.0853i$
Analytic conductor: \(21.0631\)
Root analytic conductor: \(21.0631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (1:\ ),\ 0.996 - 0.0853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.276442823 - 0.05459193518i\)
\(L(\frac12)\) \(\approx\) \(1.276442823 - 0.05459193518i\)
\(L(1)\) \(\approx\) \(0.8690025398 - 0.1385522120i\)
\(L(1)\) \(\approx\) \(0.8690025398 - 0.1385522120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
11 \( 1 + (-0.365 + 0.930i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.733 + 0.680i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.955 + 0.294i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.988 - 0.149i)T \)
53 \( 1 + (0.955 - 0.294i)T \)
59 \( 1 + (-0.0747 - 0.997i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.988 - 0.149i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.365 + 0.930i)T \)
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

−26.93409699541458351930196020896, −26.06174045935363360237728695627, −24.859009534062802236306902686405, −23.69231164900046611091392937659, −22.725512447828793265867799905195, −22.24598619759568847244593677623, −21.20313412873088204409498056901, −20.29565043738167791076876203235, −18.64704537975250987530283022051, −18.27171186236908024391963587398, −17.09282164047353883968063184413, −16.03761747273386668819983681926, −15.29409215811757633414693951351, −14.16660013511240982904702606654, −13.01509660173145484154091978317, −11.626486234794137434820876098375, −10.86469089361055188553086962142, −10.17960170584714991489680163447, −8.84035182980653444223893880467, −7.36959500372648600073734390309, −6.20581243994193851073015799577, −5.413079092420108388112345203106, −3.88496183092765564692391349352, −2.81094346703827457547093678851, −0.6604713196202959750489026911, 0.955331453190650805810973605483, 2.10265775059842681071561978645, 4.301144490947930005828247638, 5.15173797457388261376056705211, 6.34700740504918092925804481528, 7.41961716330133724559004419453, 8.640487004332279422049276168469, 9.78624864460257684413840575535, 11.12933407554217858492676844827, 11.97383010386133952710254908418, 13.02414581188777592179594156451, 13.56276968384829498966526729854, 15.34534991519202414502712717151, 16.20590539591833531426242683144, 17.26862453922644688766375627998, 17.80645950330346609179635473768, 19.039935284646044397570502465700, 19.99848927937038870127785281782, 21.09018526861615843924448505033, 22.005018218025896742242168935, 23.261323931804525185556399446062, 23.76714826146620431427324575881, 24.70177402104853069886545047548, 25.57397321561474583131166160010, 26.82013228327676088735921719921

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