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Properties

Label 1-1588-1588.1035-r0-0-0
Degree $1$
Conductor $1588$
Sign $0.396 - 0.918i$
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  + (0.902 − 0.429i)3-s + (−0.934 + 0.356i)5-s + (0.776 − 0.630i)7-s + (0.630 − 0.776i)9-s + (−0.678 + 0.734i)11-s + (−0.666 − 0.745i)13-s + (−0.690 + 0.723i)15-s + (−0.189 + 0.981i)17-s + (0.916 + 0.400i)19-s + (0.429 − 0.902i)21-s + (−0.975 − 0.220i)23-s + (0.745 − 0.666i)25-s + (0.235 − 0.971i)27-s + (0.873 + 0.486i)29-s + (−0.415 − 0.909i)31-s + ⋯
L(s)  = 1  + (0.902 − 0.429i)3-s + (−0.934 + 0.356i)5-s + (0.776 − 0.630i)7-s + (0.630 − 0.776i)9-s + (−0.678 + 0.734i)11-s + (−0.666 − 0.745i)13-s + (−0.690 + 0.723i)15-s + (−0.189 + 0.981i)17-s + (0.916 + 0.400i)19-s + (0.429 − 0.902i)21-s + (−0.975 − 0.220i)23-s + (0.745 − 0.666i)25-s + (0.235 − 0.971i)27-s + (0.873 + 0.486i)29-s + (−0.415 − 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $0.396 - 0.918i$
Analytic conductor: \(7.37464\)
Root analytic conductor: \(7.37464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1588} (1035, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (0:\ ),\ 0.396 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.515490886 - 0.9967160344i\)
\(L(\frac12)\) \(\approx\) \(1.515490886 - 0.9967160344i\)
\(L(1)\) \(\approx\) \(1.243339685 - 0.2945691625i\)
\(L(1)\) \(\approx\) \(1.243339685 - 0.2945691625i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 + (0.902 - 0.429i)T \)
5 \( 1 + (-0.934 + 0.356i)T \)
7 \( 1 + (0.776 - 0.630i)T \)
11 \( 1 + (-0.678 + 0.734i)T \)
13 \( 1 + (-0.666 - 0.745i)T \)
17 \( 1 + (-0.189 + 0.981i)T \)
19 \( 1 + (0.916 + 0.400i)T \)
23 \( 1 + (-0.975 - 0.220i)T \)
29 \( 1 + (0.873 + 0.486i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (0.967 - 0.251i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (0.981 - 0.189i)T \)
47 \( 1 + (-0.527 - 0.849i)T \)
53 \( 1 + (0.690 + 0.723i)T \)
59 \( 1 + (-0.567 - 0.823i)T \)
61 \( 1 + (0.666 - 0.745i)T \)
67 \( 1 + (0.701 + 0.712i)T \)
71 \( 1 + (0.0950 - 0.995i)T \)
73 \( 1 + (0.997 - 0.0634i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.327 + 0.945i)T \)
89 \( 1 + (-0.978 - 0.204i)T \)
97 \( 1 + (-0.857 - 0.513i)T \)
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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

−20.606294414522249112595183003323, −19.84573794284805491245316239009, −19.35007093280329374789641884994, −18.46143159673317899118694349104, −17.90806938171479464930426979651, −16.51858504729826658315388096234, −15.98572855568832680205079926661, −15.53611305949834186507898157215, −14.56996445389279711828481904202, −14.062642911045381021300872611561, −13.22925345296086391124046269967, −12.17246947641612886792043599132, −11.54905334800557847958074139063, −10.86654206043544748863228541087, −9.65468420564759837582593005254, −9.11977913377786349444651879796, −8.16485935185266342063553715714, −7.86590453812435016528687388091, −6.93890861968300807142252468936, −5.45315290577959504064288516101, −4.78027311912368956499075219344, −4.13232665465993045411273897460, −2.97186473516455296142160277047, −2.43992107817075655847734910444, −1.11726472662874277083666464268, 0.6648994379857213247974669743, 1.915596597768664068988802235789, 2.694836052761747356282714614625, 3.7629208577794064321663188201, 4.29350339727646413891587002340, 5.37371779039899816968571608344, 6.67059628102941157449868894274, 7.59794567007384739562990970659, 7.780036620024722046052478517757, 8.47892947298658292241864001329, 9.74836552281138466378166335817, 10.3808204033558884018205455393, 11.2037907307740140937496569017, 12.337424314581155810004931720050, 12.60024482388970005387518463834, 13.760886601305040612693726698794, 14.38721128162168173231306866418, 15.08676143854361653269514517143, 15.53942634480140114546153330352, 16.56417843394001929647249362816, 17.73750297723016836889922887585, 18.07322303253031158473419538828, 18.93402554620443275084789406041, 19.877477341150408703528559420202, 20.119731956841106124511559975732

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