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Properties

Label 1-1588-1588.1263-r0-0-0
Degree $1$
Conductor $1588$
Sign $-0.940 - 0.339i$
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  + (0.527 − 0.849i)3-s + (0.0634 − 0.997i)5-s + (−0.895 − 0.444i)7-s + (−0.444 − 0.895i)9-s + (0.857 − 0.513i)11-s + (0.126 − 0.991i)13-s + (−0.814 − 0.580i)15-s + (0.971 − 0.235i)17-s + (0.266 − 0.963i)19-s + (−0.849 + 0.527i)21-s + (0.873 + 0.486i)23-s + (−0.991 − 0.126i)25-s + (−0.995 − 0.0950i)27-s + (0.805 − 0.592i)29-s + (0.142 − 0.989i)31-s + ⋯
L(s)  = 1  + (0.527 − 0.849i)3-s + (0.0634 − 0.997i)5-s + (−0.895 − 0.444i)7-s + (−0.444 − 0.895i)9-s + (0.857 − 0.513i)11-s + (0.126 − 0.991i)13-s + (−0.814 − 0.580i)15-s + (0.971 − 0.235i)17-s + (0.266 − 0.963i)19-s + (−0.849 + 0.527i)21-s + (0.873 + 0.486i)23-s + (−0.991 − 0.126i)25-s + (−0.995 − 0.0950i)27-s + (0.805 − 0.592i)29-s + (0.142 − 0.989i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $-0.940 - 0.339i$
Analytic conductor: \(7.37464\)
Root analytic conductor: \(7.37464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1588} (1263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (0:\ ),\ -0.940 - 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3289612465 - 1.881224283i\)
\(L(\frac12)\) \(\approx\) \(0.3289612465 - 1.881224283i\)
\(L(1)\) \(\approx\) \(0.9670988479 - 0.8493343861i\)
\(L(1)\) \(\approx\) \(0.9670988479 - 0.8493343861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 + (0.527 - 0.849i)T \)
5 \( 1 + (0.0634 - 0.997i)T \)
7 \( 1 + (-0.895 - 0.444i)T \)
11 \( 1 + (0.857 - 0.513i)T \)
13 \( 1 + (0.126 - 0.991i)T \)
17 \( 1 + (0.971 - 0.235i)T \)
19 \( 1 + (0.266 - 0.963i)T \)
23 \( 1 + (0.873 + 0.486i)T \)
29 \( 1 + (0.805 - 0.592i)T \)
31 \( 1 + (0.142 - 0.989i)T \)
37 \( 1 + (0.950 + 0.312i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (0.235 - 0.971i)T \)
47 \( 1 + (-0.296 + 0.954i)T \)
53 \( 1 + (0.814 - 0.580i)T \)
59 \( 1 + (0.0317 + 0.999i)T \)
61 \( 1 + (-0.126 - 0.991i)T \)
67 \( 1 + (0.204 + 0.978i)T \)
71 \( 1 + (-0.618 + 0.786i)T \)
73 \( 1 + (-0.0792 + 0.996i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (0.723 + 0.690i)T \)
89 \( 1 + (-0.795 + 0.605i)T \)
97 \( 1 + (0.110 - 0.993i)T \)
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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

−21.04184818704368398208117006691, −19.93067476096505174917130345044, −19.33160657714905221439199940972, −18.81961626704217308653418135292, −17.97593079042492214335986173656, −16.73760235295523678981248521700, −16.42245584783574910936900126994, −15.489624472127966911272677570049, −14.68366140671399624979891136287, −14.36828972796085794082286029275, −13.55893601695600948298141926567, −12.359594662088501387653920508751, −11.76145239934063180729578607335, −10.67306654963832906996608391854, −10.11799564990322824615650982081, −9.389783448079085512254962137323, −8.809403319442964979465083854975, −7.69124729636135446622242885151, −6.77272198966170384955221426865, −6.14706851977774487398315638655, −5.087706386216128830256308023995, −3.981123892730391101951207699399, −3.39474195831597737997159532602, −2.65043770361883358224715818304, −1.615136051682836973684944711267, 0.81600148544771342590998812317, 1.05207476211206032830244718154, 2.61790771988829656499596709121, 3.312706918067317732352256148171, 4.20934153391392911769145665311, 5.45135469219189398003427464584, 6.15697763773124319635140752263, 7.02622653391640831723555267317, 7.85504407548612174144682877032, 8.55763664634317728547778754359, 9.40961436196192576244509184380, 9.87708343540860790003638343115, 11.27646430941104912152760027654, 11.99750400727395326195116046186, 12.79660997029349144766135356430, 13.34926361559845356694560637518, 13.81412168334544509683616209296, 14.8849820038242980848857529470, 15.6762621404521522832917355674, 16.590914008658671230214006692880, 17.16721207128294067254198952737, 17.85341833203853719951786834190, 18.95775706044376634384554680706, 19.43474136082140418401426288875, 20.09101997038066561865174817327

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