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Properties

Label 1-1588-1588.1107-r0-0-0
Degree $1$
Conductor $1588$
Sign $0.927 - 0.373i$
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  + (−0.204 − 0.978i)3-s + (0.993 + 0.110i)5-s + (−0.400 + 0.916i)7-s + (−0.916 + 0.400i)9-s + (0.987 − 0.158i)11-s + (0.220 − 0.975i)13-s + (−0.0950 − 0.995i)15-s + (−0.690 + 0.723i)17-s + (0.0792 − 0.996i)19-s + (0.978 + 0.204i)21-s + (0.630 + 0.776i)23-s + (0.975 + 0.220i)25-s + (0.580 + 0.814i)27-s + (−0.444 + 0.895i)29-s + (0.142 + 0.989i)31-s + ⋯
L(s)  = 1  + (−0.204 − 0.978i)3-s + (0.993 + 0.110i)5-s + (−0.400 + 0.916i)7-s + (−0.916 + 0.400i)9-s + (0.987 − 0.158i)11-s + (0.220 − 0.975i)13-s + (−0.0950 − 0.995i)15-s + (−0.690 + 0.723i)17-s + (0.0792 − 0.996i)19-s + (0.978 + 0.204i)21-s + (0.630 + 0.776i)23-s + (0.975 + 0.220i)25-s + (0.580 + 0.814i)27-s + (−0.444 + 0.895i)29-s + (0.142 + 0.989i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(7.37464\)
Root analytic conductor: \(7.37464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1588} (1107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (0:\ ),\ 0.927 - 0.373i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.730622686 - 0.3355548915i\)
\(L(\frac12)\) \(\approx\) \(1.730622686 - 0.3355548915i\)
\(L(1)\) \(\approx\) \(1.178363781 - 0.2125815522i\)
\(L(1)\) \(\approx\) \(1.178363781 - 0.2125815522i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 + (-0.204 - 0.978i)T \)
5 \( 1 + (0.993 + 0.110i)T \)
7 \( 1 + (-0.400 + 0.916i)T \)
11 \( 1 + (0.987 - 0.158i)T \)
13 \( 1 + (0.220 - 0.975i)T \)
17 \( 1 + (-0.690 + 0.723i)T \)
19 \( 1 + (0.0792 - 0.996i)T \)
23 \( 1 + (0.630 + 0.776i)T \)
29 \( 1 + (-0.444 + 0.895i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (0.527 - 0.849i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (0.723 - 0.690i)T \)
47 \( 1 + (0.605 + 0.795i)T \)
53 \( 1 + (0.0950 - 0.995i)T \)
59 \( 1 + (0.666 - 0.745i)T \)
61 \( 1 + (-0.220 - 0.975i)T \)
67 \( 1 + (0.999 + 0.0317i)T \)
71 \( 1 + (-0.371 + 0.928i)T \)
73 \( 1 + (0.967 - 0.251i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.235 + 0.971i)T \)
89 \( 1 + (-0.734 + 0.678i)T \)
97 \( 1 + (-0.553 + 0.832i)T \)
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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

−20.61419052836344178518189125370, −20.11484384981996560536074079458, −19.06550382427449268037373270591, −18.21615224213867608983488872708, −17.15827006665228634720587552284, −16.82376329467013231067552479245, −16.38696359299161524117988250825, −15.26731764528320515833970065020, −14.46097630420012582920219457468, −13.85491071565114880853808686975, −13.23816506211202860829883126660, −12.036098198694971337500811674866, −11.32949237930904978672447057348, −10.49047150703781098696910609153, −9.75598812782426138309570367921, −9.32107674944654687627511637806, −8.540541167672434348397639777270, −7.14091372362379056550706823499, −6.39101261844110272727585496675, −5.79292438638837825350684725060, −4.50144613645480390239608998922, −4.22160926426433520693949998092, −3.12102065095884609979602358663, −2.0276215543802617759604051676, −0.84591190086733824129555522160, 0.95894717403880032696104992039, 1.87625840057332893653396209376, 2.667026671149974485851244697853, 3.52012302005517815477278060014, 5.21017537735855702533902942619, 5.604242503007454408200311162160, 6.591891478301950904699738334696, 6.88127186005166729101548302587, 8.23755810427813694372721242827, 8.94195216484828889512644783599, 9.50963821869919068922561585420, 10.7811881896978994250908838437, 11.281398401306947305402066210784, 12.413479727443420477776020634543, 12.84557624862217380889262605528, 13.51553354878716687755878049616, 14.33164227394739291042262570114, 15.09937953136437116322458317431, 15.97941651611991217156819253411, 17.0964295033237125811936540586, 17.54163778073696635342681227527, 18.1026370228855481112145100517, 18.9132552345698301962671404229, 19.61562695235037675442620359774, 20.209739823604345511318515696865

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