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Properties

Label 1-1588-1588.1211-r0-0-0
Degree $1$
Conductor $1588$
Sign $0.000520 + 0.999i$
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  + (−0.444 + 0.895i)3-s + (0.126 + 0.991i)5-s + (0.795 + 0.605i)7-s + (−0.605 − 0.795i)9-s + (−0.472 − 0.881i)11-s + (0.251 + 0.967i)13-s + (−0.945 − 0.327i)15-s + (0.458 − 0.888i)17-s + (0.857 − 0.513i)19-s + (−0.895 + 0.444i)21-s + (0.527 − 0.849i)23-s + (−0.967 + 0.251i)25-s + (0.981 − 0.189i)27-s + (0.296 + 0.954i)29-s + (0.959 − 0.281i)31-s + ⋯
L(s)  = 1  + (−0.444 + 0.895i)3-s + (0.126 + 0.991i)5-s + (0.795 + 0.605i)7-s + (−0.605 − 0.795i)9-s + (−0.472 − 0.881i)11-s + (0.251 + 0.967i)13-s + (−0.945 − 0.327i)15-s + (0.458 − 0.888i)17-s + (0.857 − 0.513i)19-s + (−0.895 + 0.444i)21-s + (0.527 − 0.849i)23-s + (−0.967 + 0.251i)25-s + (0.981 − 0.189i)27-s + (0.296 + 0.954i)29-s + (0.959 − 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000520 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000520 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $0.000520 + 0.999i$
Analytic conductor: \(7.37464\)
Root analytic conductor: \(7.37464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1588} (1211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (0:\ ),\ 0.000520 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126055493 + 1.125469543i\)
\(L(\frac12)\) \(\approx\) \(1.126055493 + 1.125469543i\)
\(L(1)\) \(\approx\) \(0.9769079036 + 0.4898462512i\)
\(L(1)\) \(\approx\) \(0.9769079036 + 0.4898462512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 + (-0.444 + 0.895i)T \)
5 \( 1 + (0.126 + 0.991i)T \)
7 \( 1 + (0.795 + 0.605i)T \)
11 \( 1 + (-0.472 - 0.881i)T \)
13 \( 1 + (0.251 + 0.967i)T \)
17 \( 1 + (0.458 - 0.888i)T \)
19 \( 1 + (0.857 - 0.513i)T \)
23 \( 1 + (0.527 - 0.849i)T \)
29 \( 1 + (0.296 + 0.954i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (0.805 - 0.592i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (-0.888 + 0.458i)T \)
47 \( 1 + (0.823 - 0.567i)T \)
53 \( 1 + (0.945 - 0.327i)T \)
59 \( 1 + (-0.0634 + 0.997i)T \)
61 \( 1 + (-0.251 + 0.967i)T \)
67 \( 1 + (0.916 + 0.400i)T \)
71 \( 1 + (0.971 + 0.235i)T \)
73 \( 1 + (-0.987 + 0.158i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.0475 - 0.998i)T \)
89 \( 1 + (0.963 - 0.266i)T \)
97 \( 1 + (-0.975 + 0.220i)T \)
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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

−20.29873087512567082557479261848, −19.68551015047547327098105845759, −18.70358896513816243229633363178, −17.89075179464737300720355568602, −17.27493322296189583059667043160, −17.01122149273158296019351939486, −15.8538724999098844198729795384, −15.1358086985442712998141072831, −13.99191529218605605711400883992, −13.45653577204055615105129218550, −12.68828585569174498031015810319, −12.14968954102423052553337269952, −11.33412733637414321307234074970, −10.414362740076602382803930615769, −9.72677477806889736054104674942, −8.39232619698270661984474012072, −7.91309576486150156628854205615, −7.37848789959932925690928195936, −6.161716680569778127989244638449, −5.36085902469959889380972283607, −4.83978315954267554394409492581, −3.74421505965852588248037957312, −2.38443803517332118304949968893, −1.40848266688826210539592550822, −0.85043287845321026946953344255, 0.98666963865055724826703902974, 2.57562058623164278023956605610, 3.02837179748989990742950193729, 4.18639751782783770382067231676, 5.04584523704793924063701831107, 5.74113594686861675015491469728, 6.55654356039397997299464363663, 7.47801766192941760145048216423, 8.57978945966553167119075955913, 9.2115962293839219880440395244, 10.13047256313246734693988624213, 10.86078081892256398953701374374, 11.56385464817152332362973660393, 11.82637557394342021359569108247, 13.32493092588255755219914007506, 14.24563648478110598599483939766, 14.591684779849697832843372730575, 15.50401255936994313396869128856, 16.16172958580345793209636604161, 16.796441379729838526592808693093, 17.92949241741158655218981718482, 18.29652448103866102189074940880, 18.949630564808642036175568186788, 20.07697497002302634601651352344, 20.99665353570970024502704908579

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