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Properties

Label 1-1588-1588.1079-r0-0-0
Degree $1$
Conductor $1588$
Sign $-0.978 - 0.206i$
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  + (−0.916 + 0.400i)3-s + (0.220 − 0.975i)5-s + (0.734 − 0.678i)7-s + (0.678 − 0.734i)9-s + (−0.950 + 0.312i)11-s + (−0.429 + 0.902i)13-s + (0.189 + 0.981i)15-s + (−0.998 + 0.0475i)17-s + (0.987 + 0.158i)19-s + (−0.400 + 0.916i)21-s + (−0.204 + 0.978i)23-s + (−0.902 − 0.429i)25-s + (−0.327 + 0.945i)27-s + (−0.605 − 0.795i)29-s + (0.959 − 0.281i)31-s + ⋯
L(s)  = 1  + (−0.916 + 0.400i)3-s + (0.220 − 0.975i)5-s + (0.734 − 0.678i)7-s + (0.678 − 0.734i)9-s + (−0.950 + 0.312i)11-s + (−0.429 + 0.902i)13-s + (0.189 + 0.981i)15-s + (−0.998 + 0.0475i)17-s + (0.987 + 0.158i)19-s + (−0.400 + 0.916i)21-s + (−0.204 + 0.978i)23-s + (−0.902 − 0.429i)25-s + (−0.327 + 0.945i)27-s + (−0.605 − 0.795i)29-s + (0.959 − 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $-0.978 - 0.206i$
Analytic conductor: \(7.37464\)
Root analytic conductor: \(7.37464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1588} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (0:\ ),\ -0.978 - 0.206i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03410535670 - 0.3266377844i\)
\(L(\frac12)\) \(\approx\) \(0.03410535670 - 0.3266377844i\)
\(L(1)\) \(\approx\) \(0.6865445443 - 0.1144713582i\)
\(L(1)\) \(\approx\) \(0.6865445443 - 0.1144713582i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 + (-0.916 + 0.400i)T \)
5 \( 1 + (0.220 - 0.975i)T \)
7 \( 1 + (0.734 - 0.678i)T \)
11 \( 1 + (-0.950 + 0.312i)T \)
13 \( 1 + (-0.429 + 0.902i)T \)
17 \( 1 + (-0.998 + 0.0475i)T \)
19 \( 1 + (0.987 + 0.158i)T \)
23 \( 1 + (-0.204 + 0.978i)T \)
29 \( 1 + (-0.605 - 0.795i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (-0.444 - 0.895i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (0.0475 - 0.998i)T \)
47 \( 1 + (0.266 - 0.963i)T \)
53 \( 1 + (-0.189 + 0.981i)T \)
59 \( 1 + (-0.993 + 0.110i)T \)
61 \( 1 + (0.429 + 0.902i)T \)
67 \( 1 + (-0.997 - 0.0634i)T \)
71 \( 1 + (-0.690 + 0.723i)T \)
73 \( 1 + (0.873 - 0.486i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.888 + 0.458i)T \)
89 \( 1 + (-0.996 - 0.0792i)T \)
97 \( 1 + (-0.386 - 0.922i)T \)
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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

−20.995625449343091878227931147538, −20.08944123210781053550373857341, −19.00743176621770338356568218617, −18.415902466177026340426082272548, −17.882227732965944815337611551719, −17.51544984545042189796612994698, −16.324763870529336121058580856425, −15.57325306629186838262377697949, −14.977725423826128332243403261408, −14.01575130831245054954109929667, −13.24999230580557691848382849754, −12.471644814715437679094103475265, −11.620673582260143494833917913325, −10.97563734145523473887007169284, −10.45817767229110833846291287697, −9.57142051683305777657529571164, −8.23007291346351625473489956879, −7.71944001713868363902505039699, −6.75726024680646902114233407258, −6.06341231431130549265823333358, −5.20915626957677901017603617737, −4.6934906677339419165665494484, −3.044469540508364737534369303705, −2.46413411429362400033327713042, −1.389593843732310658910190390062, 0.13868622908274276780761905248, 1.34287105620247213133295387993, 2.19266384675569326019171227526, 3.91200922390326352582815506099, 4.44874675791145610397783381511, 5.215656053708714535566995425230, 5.76648447794048427826440496735, 7.04752627180991586170914576831, 7.612429707870261004924840220502, 8.72816764350776777713128051454, 9.56093243024231458453862194128, 10.208765386718408334701653256437, 11.0757634601508437064592988881, 11.8032965365858835731382216453, 12.37563436747610735062492966927, 13.51592391486216071019545147318, 13.80562422743305662111988512206, 15.240237584335802433737897645593, 15.69506756303344610117722185513, 16.56254461641277826887618388612, 17.178885464400325378688357264332, 17.67559830238205980074645446550, 18.38642014547890275672749083209, 19.51578479481912770996536140762, 20.43600806333002986314659175800

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