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Properties

Label 1-388-388.271-r0-0-0
Degree $1$
Conductor $388$
Sign $-0.0901 + 0.995i$
Analytic cond. $1.80186$
Root an. cond. $1.80186$
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  + (0.382 − 0.923i)3-s + (0.195 + 0.980i)5-s + (−0.195 + 0.980i)7-s + (−0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.980 + 0.195i)13-s + (0.980 + 0.195i)15-s + (−0.980 + 0.195i)17-s + (0.195 + 0.980i)19-s + (0.831 + 0.555i)21-s + (−0.555 + 0.831i)23-s + (−0.923 + 0.382i)25-s + (−0.923 + 0.382i)27-s + (0.555 − 0.831i)29-s + (−0.923 − 0.382i)31-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (0.195 + 0.980i)5-s + (−0.195 + 0.980i)7-s + (−0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.980 + 0.195i)13-s + (0.980 + 0.195i)15-s + (−0.980 + 0.195i)17-s + (0.195 + 0.980i)19-s + (0.831 + 0.555i)21-s + (−0.555 + 0.831i)23-s + (−0.923 + 0.382i)25-s + (−0.923 + 0.382i)27-s + (0.555 − 0.831i)29-s + (−0.923 − 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0901 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0901 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(388\)    =    \(2^{2} \cdot 97\)
Sign: $-0.0901 + 0.995i$
Analytic conductor: \(1.80186\)
Root analytic conductor: \(1.80186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{388} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 388,\ (0:\ ),\ -0.0901 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6291276935 + 0.6886431014i\)
\(L(\frac12)\) \(\approx\) \(0.6291276935 + 0.6886431014i\)
\(L(1)\) \(\approx\) \(0.9420781081 + 0.1733507741i\)
\(L(1)\) \(\approx\) \(0.9420781081 + 0.1733507741i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 \)
good3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (0.195 + 0.980i)T \)
7 \( 1 + (-0.195 + 0.980i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.980 + 0.195i)T \)
17 \( 1 + (-0.980 + 0.195i)T \)
19 \( 1 + (0.195 + 0.980i)T \)
23 \( 1 + (-0.555 + 0.831i)T \)
29 \( 1 + (0.555 - 0.831i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (0.831 - 0.555i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (0.382 + 0.923i)T \)
59 \( 1 + (0.555 + 0.831i)T \)
61 \( 1 + T \)
67 \( 1 + (0.980 - 0.195i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (0.195 + 0.980i)T \)
89 \( 1 + (0.382 - 0.923i)T \)
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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

−24.19866571606375665713288010899, −23.56977517368581469511971339053, −22.07175409712116062895450760424, −21.83082453599187384544278123672, −20.51846409353149471002021442551, −20.13489028348203704862414207745, −19.428118235257027125246496792686, −17.88614441883844225573276414119, −16.91843324796729311884984223379, −16.35082959240510922502156791429, −15.60285231909760042240655659793, −14.39807158908781464903842155612, −13.59028619248353945707762214694, −12.85909222537806216954045018711, −11.46167443628972112771976046837, −10.54314041807732490349502629581, −9.7129752194232447996036214084, −8.80242782110979723279333165385, −8.02058769431832424796986757458, −6.70003961674780873370556603159, −5.20790406759109540758725150164, −4.64748352559653251342947861558, −3.54092705061916560620823502048, −2.34665920957616451640138051815, −0.48301262696906484646650810367, 2.106818699116495737197195915552, 2.362584204631810951073214777433, 3.7448823777639440408186450503, 5.4242128173934726379679576289, 6.36491155044691701752072382814, 7.24018646468191619309723186005, 8.040012786920494436241344966631, 9.3126130162844874938258380568, 10.06575772016383556944916423102, 11.483306704919714783785161928355, 12.19735028736416448859005468646, 13.07640269787937982212331115547, 14.08877239799672261228977317884, 14.939506709636343395010317424016, 15.485283545049792823252375167177, 17.093865381626688887095493465815, 18.05032174934118094335237165779, 18.43005857406750141389852502397, 19.40450908224539164252453606672, 20.07812842601886846089082449317, 21.35853253009112716686683013628, 22.19043474734410007091350876592, 22.9630059487220897335956803783, 23.8892813557607219306428226924, 24.935974902265584866442678904847

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