↑

Properties

Label 1-1580-1580.439-r1-0-0
Degree $1$
Conductor $1580$
Sign $-0.968 - 0.250i$
Analytic cond. $169.794$
Root an. cond. $169.794$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 − 0.160i)3-s + (−0.632 − 0.774i)7-s + (0.948 − 0.316i)9-s + (0.0402 − 0.999i)11-s + (−0.692 − 0.721i)13-s + (0.970 − 0.239i)17-s + (−0.428 − 0.903i)19-s + (−0.748 − 0.663i)21-s + (−0.5 + 0.866i)23-s + (0.885 − 0.464i)27-s + (−0.200 − 0.979i)29-s + (0.919 + 0.391i)31-s + (−0.120 − 0.992i)33-s + (0.996 + 0.0804i)37-s + (−0.799 − 0.600i)39-s + ⋯
L(s)  = 1  + (0.987 − 0.160i)3-s + (−0.632 − 0.774i)7-s + (0.948 − 0.316i)9-s + (0.0402 − 0.999i)11-s + (−0.692 − 0.721i)13-s + (0.970 − 0.239i)17-s + (−0.428 − 0.903i)19-s + (−0.748 − 0.663i)21-s + (−0.5 + 0.866i)23-s + (0.885 − 0.464i)27-s + (−0.200 − 0.979i)29-s + (0.919 + 0.391i)31-s + (−0.120 − 0.992i)33-s + (0.996 + 0.0804i)37-s + (−0.799 − 0.600i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1580\)    =    \(2^{2} \cdot 5 \cdot 79\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(169.794\)
Root analytic conductor: \(169.794\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1580} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1580,\ (1:\ ),\ -0.968 - 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2638120141 - 2.076575921i\)
\(L(\frac12)\) \(\approx\) \(0.2638120141 - 2.076575921i\)
\(L(1)\) \(\approx\) \(1.211411924 - 0.5262628992i\)
\(L(1)\) \(\approx\) \(1.211411924 - 0.5262628992i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
79 \( 1 \)
good3 \( 1 + (0.987 - 0.160i)T \)
7 \( 1 + (-0.632 - 0.774i)T \)
11 \( 1 + (0.0402 - 0.999i)T \)
13 \( 1 + (-0.692 - 0.721i)T \)
17 \( 1 + (0.970 - 0.239i)T \)
19 \( 1 + (-0.428 - 0.903i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.200 - 0.979i)T \)
31 \( 1 + (0.919 + 0.391i)T \)
37 \( 1 + (0.996 + 0.0804i)T \)
41 \( 1 + (0.885 + 0.464i)T \)
43 \( 1 + (-0.0402 - 0.999i)T \)
47 \( 1 + (-0.996 + 0.0804i)T \)
53 \( 1 + (-0.987 - 0.160i)T \)
59 \( 1 + (-0.278 - 0.960i)T \)
61 \( 1 + (0.568 - 0.822i)T \)
67 \( 1 + (0.120 - 0.992i)T \)
71 \( 1 + (0.354 + 0.935i)T \)
73 \( 1 + (-0.692 + 0.721i)T \)
83 \( 1 + (0.278 - 0.960i)T \)
89 \( 1 + (-0.354 + 0.935i)T \)
97 \( 1 + (-0.568 + 0.822i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.754645797963362468168115298681, −19.77847540044405712032347908393, −19.269028701852800654972187958728, −18.61035071584319711031867200515, −17.89373808584348680712297615491, −16.612461263625996443281554378195, −16.25435524247855690079668393458, −15.190238767512836886497810445803, −14.6565547826351232650362158853, −14.18503054493904605537242649102, −12.9273016952760790126567598925, −12.52341335030011647707443183627, −11.814668934858696973192361596035, −10.416452959189843762214108322889, −9.78957662732463715967821186269, −9.296620234430920421925032858759, −8.35124267983850947636411658693, −7.64948481553176562339761946049, −6.76214929860052496144624309307, −5.90191114244835451674603718356, −4.71209603007589186859365527960, −4.054326907024001057356019589755, −2.98634825303401919570692692156, −2.30276933191952181817505821135, −1.44693389361617880237660169794, 0.332510194297569999174507649925, 1.133121552603336651206069601055, 2.49636295737716690913136273665, 3.17825456475220919753728094156, 3.86660653524258523937488520034, 4.89844905879664137162419236835, 6.05160804972752476214894981356, 6.85873209787582965285423902245, 7.81132995363029661784978616080, 8.15007352475369236941969387316, 9.42940430823516691038447710849, 9.78107453065843415519104935176, 10.67696271691261613443362403555, 11.656345079299969007107548171957, 12.68997317509217010916874136543, 13.26884521170405162012000488401, 13.92920001283596250229795136117, 14.56013148510342631150460962143, 15.53230900842934681991271751668, 16.06564280322895909754370465358, 17.03661781012515039986434115602, 17.69443146337566806330631616644, 18.833878719616935959609265721, 19.29170633030063940649854291062, 19.88739945450128009862650661601

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