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Properties

Label 1-1588-1588.135-r0-0-0
Degree $1$
Conductor $1588$
Sign $0.614 - 0.788i$
Analytic cond. $7.37464$
Root an. cond. $7.37464$
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  + (0.805 − 0.592i)3-s + (−0.734 + 0.678i)5-s + (0.954 − 0.296i)7-s + (0.296 − 0.954i)9-s + (0.745 − 0.666i)11-s + (−0.996 − 0.0792i)13-s + (−0.189 + 0.981i)15-s + (0.998 + 0.0475i)17-s + (−0.356 + 0.934i)19-s + (0.592 − 0.805i)21-s + (0.950 + 0.312i)23-s + (0.0792 − 0.996i)25-s + (−0.327 − 0.945i)27-s + (−0.386 − 0.922i)29-s + (0.959 + 0.281i)31-s + ⋯
L(s)  = 1  + (0.805 − 0.592i)3-s + (−0.734 + 0.678i)5-s + (0.954 − 0.296i)7-s + (0.296 − 0.954i)9-s + (0.745 − 0.666i)11-s + (−0.996 − 0.0792i)13-s + (−0.189 + 0.981i)15-s + (0.998 + 0.0475i)17-s + (−0.356 + 0.934i)19-s + (0.592 − 0.805i)21-s + (0.950 + 0.312i)23-s + (0.0792 − 0.996i)25-s + (−0.327 − 0.945i)27-s + (−0.386 − 0.922i)29-s + (0.959 + 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $0.614 - 0.788i$
Analytic conductor: \(7.37464\)
Root analytic conductor: \(7.37464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1588} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (0:\ ),\ 0.614 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.952773821 - 0.9536710119i\)
\(L(\frac12)\) \(\approx\) \(1.952773821 - 0.9536710119i\)
\(L(1)\) \(\approx\) \(1.390872836 - 0.3124180071i\)
\(L(1)\) \(\approx\) \(1.390872836 - 0.3124180071i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 + (0.805 - 0.592i)T \)
5 \( 1 + (-0.734 + 0.678i)T \)
7 \( 1 + (0.954 - 0.296i)T \)
11 \( 1 + (0.745 - 0.666i)T \)
13 \( 1 + (-0.996 - 0.0792i)T \)
17 \( 1 + (0.998 + 0.0475i)T \)
19 \( 1 + (-0.356 + 0.934i)T \)
23 \( 1 + (0.950 + 0.312i)T \)
29 \( 1 + (-0.386 - 0.922i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (-0.553 - 0.832i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (0.0475 + 0.998i)T \)
47 \( 1 + (0.701 - 0.712i)T \)
53 \( 1 + (0.189 + 0.981i)T \)
59 \( 1 + (-0.400 - 0.916i)T \)
61 \( 1 + (0.996 - 0.0792i)T \)
67 \( 1 + (0.444 - 0.895i)T \)
71 \( 1 + (0.690 + 0.723i)T \)
73 \( 1 + (-0.857 + 0.513i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.888 - 0.458i)T \)
89 \( 1 + (-0.567 - 0.823i)T \)
97 \( 1 + (0.991 - 0.126i)T \)
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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

−20.63376293124903398196082218511, −19.93784213875255676180694568444, −19.26942347447308063768793795562, −18.64946423034506425114396633807, −17.19023023077674732171804679938, −17.07500571105592098630362296280, −15.978701693375039805916528991105, −15.07797822876391763126737631827, −14.882057977889006678776600561576, −14.05672894888727816259648533685, −13.048545005459390663281201988466, −12.16571636415458318353208072114, −11.64266703329060239117962308217, −10.639841291477321841697528714149, −9.760119985047654753563688456201, −8.91907086741561175980547760209, −8.494028932366364439099977894901, −7.54260173703267345819846617054, −6.99680659553546463747988542439, −5.23446150719852390323806013501, −4.83539260424726835750005598812, −4.13243026512606389595436979268, −3.13721600413398046952906827751, −2.14281369886743269534360862977, −1.16212305387677980539555187054, 0.836601100169322679790253203518, 1.822482784783133302872891309441, 2.85814055050972725885222376849, 3.62268401241095042828027294832, 4.35799458858965684941040894821, 5.592684676751987134316774993440, 6.66815391162120947475190946927, 7.36579868122050715177564671879, 7.9965115428321877133618118858, 8.55173939661141583225259079710, 9.66344468215133289506086543720, 10.46445820301963751275690883451, 11.50806621826055113837814276449, 11.95163314556551988548274310174, 12.78675997780760792264739041814, 13.994720305497472341279870042224, 14.31255632440202742121550784902, 14.90979101951990368589177294344, 15.616481366153835615453945890126, 16.9242686337967976917077804994, 17.32127364596113554322659036745, 18.510150669496798791030847477056, 18.90461446928635088978389145281, 19.542728431874889758146808275911, 20.208781738919411354138063382774

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