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Properties

Label 1-388-388.139-r0-0-0
Degree $1$
Conductor $388$
Sign $0.754 + 0.655i$
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  + (0.923 − 0.382i)3-s + (−0.831 + 0.555i)5-s + (0.831 + 0.555i)7-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.555 + 0.831i)13-s + (−0.555 + 0.831i)15-s + (0.555 + 0.831i)17-s + (−0.831 + 0.555i)19-s + (0.980 + 0.195i)21-s + (0.195 − 0.980i)23-s + (0.382 − 0.923i)25-s + (0.382 − 0.923i)27-s + (−0.195 + 0.980i)29-s + (0.382 + 0.923i)31-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (−0.831 + 0.555i)5-s + (0.831 + 0.555i)7-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.555 + 0.831i)13-s + (−0.555 + 0.831i)15-s + (0.555 + 0.831i)17-s + (−0.831 + 0.555i)19-s + (0.980 + 0.195i)21-s + (0.195 − 0.980i)23-s + (0.382 − 0.923i)25-s + (0.382 − 0.923i)27-s + (−0.195 + 0.980i)29-s + (0.382 + 0.923i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(388\)    =    \(2^{2} \cdot 97\)
Sign: $0.754 + 0.655i$
Analytic conductor: \(1.80186\)
Root analytic conductor: \(1.80186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{388} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 388,\ (0:\ ),\ 0.754 + 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.526378682 + 0.5704107455i\)
\(L(\frac12)\) \(\approx\) \(1.526378682 + 0.5704107455i\)
\(L(1)\) \(\approx\) \(1.315562695 + 0.1825464940i\)
\(L(1)\) \(\approx\) \(1.315562695 + 0.1825464940i\)

Euler product

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

−24.31892666078481970086992295071, −23.6593973575270654861926001488, −22.80430903368552600100497753629, −21.36332730136928391799115739115, −20.788779004702186046200375743911, −20.21967844591370632344312117790, −19.28148576525267965857547498472, −18.44892591319786604742553366903, −17.251913087708617379849658489266, −16.21822457582069366949936569809, −15.450392677332266578343325556477, −14.82463285035281755458794689024, −13.499631499581938857981396587779, −13.15343508817472374544785225865, −11.64116282587349477140256404118, −10.84796291664382378810489797921, −9.835133746635371110466472799523, −8.621244932278289175700319431246, −7.963988259707884573880678509036, −7.39593601095880844718045662685, −5.45972417222309976019938699558, −4.547170354239640060245622819040, −3.65452128027274543068007071475, −2.546005952474240190136019389281, −0.95356596339174961513737486517, 1.614206959136186843251075826241, 2.60722308680751722998369649881, 3.729384709130180744062169057047, 4.71759460169247527624802221587, 6.28717248098909115458376907591, 7.29292014676119189279669597538, 8.25428915561054820016230138521, 8.64370292031460561839245361183, 10.16960974816196827352154582689, 11.05284559115218782000184629336, 12.19209589528705685134287938241, 12.85088047256361711770459706547, 14.22039771538999201128539377850, 14.741706267046737085311245739911, 15.445471505057907819745448619688, 16.47829288433445523092691401782, 17.990990799265966615134747398644, 18.58874479343048418824933875545, 19.16287196503120534253392020990, 20.22030905761683309505185672171, 21.069072194413922793036288400270, 21.68727275643803166415651817500, 23.26282511707615368013690444605, 23.62859938595801010840630878221, 24.54733611783004837472470873696

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