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Properties

Label 1-14e2-196.171-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  + (−0.733 + 0.680i)3-s + (−0.955 − 0.294i)5-s + (0.0747 − 0.997i)9-s + (−0.0747 − 0.997i)11-s + (0.900 + 0.433i)13-s + (0.900 − 0.433i)15-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.623 + 0.781i)27-s + (0.623 − 0.781i)29-s + (−0.5 − 0.866i)31-s + (0.733 + 0.680i)33-s + (0.365 + 0.930i)37-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)3-s + (−0.955 − 0.294i)5-s + (0.0747 − 0.997i)9-s + (−0.0747 − 0.997i)11-s + (0.900 + 0.433i)13-s + (0.900 − 0.433i)15-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.623 + 0.781i)27-s + (0.623 − 0.781i)29-s + (−0.5 − 0.866i)31-s + (0.733 + 0.680i)33-s + (0.365 + 0.930i)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: $\chi_{196} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (0:\ ),\ 0.999 + 0.0320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7810649569 + 0.01252039841i\)
\(L(\frac12)\) \(\approx\) \(0.7810649569 + 0.01252039841i\)
\(L(1)\) \(\approx\) \(0.7774905720 + 0.05560584995i\)
\(L(1)\) \(\approx\) \(0.7774905720 + 0.05560584995i\)

Euler product

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

−27.3089216731040985965322948264, −25.84559589906542768985893292876, −25.11674602245164244350153396128, −23.80061802836651932759690486581, −23.22289711102416654104875419279, −22.658671610889717762661686932689, −21.359930358667230721596068029840, −20.07680741417519056858839479174, −19.22512842209207145353696744270, −18.304392848929257415622136109545, −17.52192382078302382171290625154, −16.32253648038283286568019901841, −15.470525570894464830371568621748, −14.32695225024755991494870925377, −12.91809287334424119784160224471, −12.293739956128563046303429670797, −11.17926069509215566971181103898, −10.47809414211707486052187035411, −8.75699618825876278143740358186, −7.519968084699059106818656572101, −6.91142099012556308082550740584, −5.5336305345686613800088054807, −4.35313018027911603178918046981, −2.84298215666739977281563144077, −1.09751896845697073222132935370, 0.907739125802142972876370152157, 3.374848949406527040636506924162, 4.1754848098204394429069479601, 5.45559381961774445082570636812, 6.468981591618891077922287061538, 8.00453811854465576521622893542, 8.93783630332889072680121055906, 10.28896024374751340964630146171, 11.262795444609364478908031607291, 11.92512863262731216312055422316, 13.11633125187896315292398391993, 14.57336680690664986943843858714, 15.53769662340758764095266370924, 16.46081822562992613947994037834, 16.91398886905792306425806125522, 18.52432451908352600667932348411, 19.14679395745262246675322708165, 20.640559037064907696452355255156, 21.16940774125989681782798846921, 22.32312569408453313218270015245, 23.40967596394666682368517666928, 23.67439846544845807274635845059, 25.062729760836482704854188359356, 26.35823155993148389881624987885, 27.15352900761456139702320668542

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