Properties

Label 1-388-388.55-r0-0-0
Degree $1$
Conductor $388$
Sign $0.0215 - 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(388\)    =    \(2^{2} \cdot 97\)
Sign: $0.0215 - 0.999i$
Analytic conductor: \(1.80186\)
Root analytic conductor: \(1.80186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{388} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 388,\ (0:\ ),\ 0.0215 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217857488 - 1.191947853i\)
\(L(\frac12)\) \(\approx\) \(1.217857488 - 1.191947853i\)
\(L(1)\) \(\approx\) \(1.283683115 - 0.5382743795i\)
\(L(1)\) \(\approx\) \(1.283683115 - 0.5382743795i\)

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

−24.7567731268550604373198097848, −24.151341560727458187269927569836, −22.58999858770022633731168767489, −21.99769840950611665663584314967, −21.26689903379134309224915567676, −20.4842612212151282190175515688, −19.23614101161576788378742949214, −18.83449088648478033033660624970, −17.8563214723721613013948474479, −16.52246239575604063149702203326, −15.834954320496938174844709101783, −14.82630526873177977776202446015, −14.09569694877596571600678746746, −13.25271263377509224102773477997, −12.416632324964754335965542532890, −10.845934386410391423028929458345, −10.066757285820347574479914126504, −9.32696634560008584530066968842, −8.44697764802639153156726360096, −7.220257682247565469363511255750, −6.21063590961981249763846792857, −5.13502868851051143730327243669, −3.73741369792109794012313420683, −2.70358917707031462455422725142, −2.03654260248727031118652848121, 0.89441858193624736406691630209, 2.375191401051811701843573068106, 3.09978556172129298016559504855, 4.564011458155837267974139303472, 5.64997566232797201849514058801, 6.962630479945424793245304977528, 7.65683911643217829645950697123, 8.85528011236537578557745055213, 9.75716972924241625697653242603, 10.23515624768697108965547924619, 11.97280176269356505777490973482, 13.11453934290218813372741514968, 13.31349144570976680352747938321, 14.23737492336821154781416673743, 15.546253455085362623844451987344, 16.08379222651129734378563124408, 17.607650478934760914603372659021, 17.89295258430761068655247699530, 19.241229856648095050478326016899, 20.00527866046174254109646246481, 20.58848421169780509695890448789, 21.46095849069221379925353694626, 22.565979557135728994011178830829, 23.484225579052901504749010625268, 24.62879210782882413171717422250

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