Properties

Label 1-388-388.239-r0-0-0
Degree $1$
Conductor $388$
Sign $0.869 - 0.494i$
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 + (−0.980 + 0.195i)5-s + (0.980 + 0.195i)7-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.195 − 0.980i)13-s + (0.195 − 0.980i)15-s + (−0.195 − 0.980i)17-s + (−0.980 + 0.195i)19-s + (−0.555 + 0.831i)21-s + (−0.831 − 0.555i)23-s + (0.923 − 0.382i)25-s + (0.923 − 0.382i)27-s + (0.831 + 0.555i)29-s + (0.923 + 0.382i)31-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (−0.980 + 0.195i)5-s + (0.980 + 0.195i)7-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.195 − 0.980i)13-s + (0.195 − 0.980i)15-s + (−0.195 − 0.980i)17-s + (−0.980 + 0.195i)19-s + (−0.555 + 0.831i)21-s + (−0.831 − 0.555i)23-s + (0.923 − 0.382i)25-s + (0.923 − 0.382i)27-s + (0.831 + 0.555i)29-s + (0.923 + 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(388\)    =    \(2^{2} \cdot 97\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(1.80186\)
Root analytic conductor: \(1.80186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{388} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 388,\ (0:\ ),\ 0.869 - 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8211889295 - 0.2171628774i\)
\(L(\frac12)\) \(\approx\) \(0.8211889295 - 0.2171628774i\)
\(L(1)\) \(\approx\) \(0.8208173584 + 0.06176949671i\)
\(L(1)\) \(\approx\) \(0.8208173584 + 0.06176949671i\)

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 \( 1 + (-0.980 + 0.195i)T \)
7 \( 1 + (0.980 + 0.195i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (-0.195 - 0.980i)T \)
17 \( 1 + (-0.195 - 0.980i)T \)
19 \( 1 + (-0.980 + 0.195i)T \)
23 \( 1 + (-0.831 - 0.555i)T \)
29 \( 1 + (0.831 + 0.555i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (-0.555 - 0.831i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (0.831 - 0.555i)T \)
61 \( 1 + T \)
67 \( 1 + (0.195 + 0.980i)T \)
71 \( 1 + (-0.555 - 0.831i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (-0.980 + 0.195i)T \)
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

−24.22025113433050938379200668302, −23.76786292836301157547098708793, −23.159207432111164848208373228835, −22.08206398571870794503552099773, −21.01664475361618769159823810074, −19.91020306573473641911719436906, −19.36780075865844389793724766417, −18.48031380405161932645001142388, −17.31090510200693432783057899037, −17.06565041361049758789065192689, −15.64649880303449597255708155911, −14.74908059441816306612151333879, −13.87175352751931015336301303090, −12.72071390973174935313076548544, −11.88811998093330812016904962101, −11.42663162717728548894359452963, −10.28259363835736221214574618052, −8.68029675229709982581296319655, −7.99677275942038810334386907326, −7.12599532782012996359250052339, −6.2363050034998910481526538816, −4.72482438669086390438125363829, −4.13601029958670398719181983097, −2.249420589154396594931453384638, −1.30703539925799943937424316421, 0.59030446005716250753888478425, 2.696093758577661184905994831671, 3.79278956251586624587072029815, 4.680849523503757052164426067697, 5.598988763700799589648731347985, 6.83689180108377153812221323985, 8.25297341349770349210437210256, 8.64976203207003080487477901843, 10.17134684770910241826077324316, 10.94281631963321420925236542784, 11.63999936706455251359934636735, 12.42303805248691087984043519493, 14.117222801984136544659525640373, 14.72111484391379555094360474629, 15.68801310875297614904870397018, 16.23702940725377278377654185843, 17.3667405769797296756111826883, 18.12602008466652928799019925893, 19.27118341686958786807576334758, 20.18699939830756171578750569161, 20.959770484651707730044983538518, 21.86141850161603898796156594399, 22.630051563010356677499097403550, 23.41743896617408817135557334142, 24.301331002381972815355174792875

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