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Properties

Label 1-4029-4029.5-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.224 - 0.974i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (0.941 + 0.335i)2-s + (0.774 + 0.632i)4-s + (−0.559 + 0.828i)5-s + (−0.915 − 0.401i)7-s + (0.517 + 0.855i)8-s + (−0.805 + 0.592i)10-s + (−0.0704 − 0.997i)11-s + (−0.160 + 0.987i)13-s + (−0.728 − 0.685i)14-s + (0.200 + 0.979i)16-s + (0.373 − 0.927i)19-s + (−0.957 + 0.287i)20-s + (0.268 − 0.963i)22-s + (0.608 − 0.793i)23-s + (−0.373 − 0.927i)25-s + (−0.482 + 0.875i)26-s + ⋯
L(s)  = 1  + (0.941 + 0.335i)2-s + (0.774 + 0.632i)4-s + (−0.559 + 0.828i)5-s + (−0.915 − 0.401i)7-s + (0.517 + 0.855i)8-s + (−0.805 + 0.592i)10-s + (−0.0704 − 0.997i)11-s + (−0.160 + 0.987i)13-s + (−0.728 − 0.685i)14-s + (0.200 + 0.979i)16-s + (0.373 − 0.927i)19-s + (−0.957 + 0.287i)20-s + (0.268 − 0.963i)22-s + (0.608 − 0.793i)23-s + (−0.373 − 0.927i)25-s + (−0.482 + 0.875i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.224 - 0.974i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.224 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2229258109 - 0.2801855983i\)
\(L(\frac12)\) \(\approx\) \(0.2229258109 - 0.2801855983i\)
\(L(1)\) \(\approx\) \(1.179106796 + 0.3708416609i\)
\(L(1)\) \(\approx\) \(1.179106796 + 0.3708416609i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.941 + 0.335i)T \)
5 \( 1 + (-0.559 + 0.828i)T \)
7 \( 1 + (-0.915 - 0.401i)T \)
11 \( 1 + (-0.0704 - 0.997i)T \)
13 \( 1 + (-0.160 + 0.987i)T \)
19 \( 1 + (0.373 - 0.927i)T \)
23 \( 1 + (0.608 - 0.793i)T \)
29 \( 1 + (-0.345 + 0.938i)T \)
31 \( 1 + (-0.455 + 0.890i)T \)
37 \( 1 + (-0.860 + 0.508i)T \)
41 \( 1 + (-0.945 - 0.326i)T \)
43 \( 1 + (-0.894 - 0.446i)T \)
47 \( 1 + (-0.600 - 0.799i)T \)
53 \( 1 + (0.875 + 0.482i)T \)
59 \( 1 + (-0.100 + 0.994i)T \)
61 \( 1 + (-0.268 - 0.963i)T \)
67 \( 1 + (-0.568 - 0.822i)T \)
71 \( 1 + (-0.592 + 0.805i)T \)
73 \( 1 + (-0.973 - 0.229i)T \)
83 \( 1 + (-0.100 - 0.994i)T \)
89 \( 1 + (-0.239 - 0.970i)T \)
97 \( 1 + (0.491 - 0.871i)T \)
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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

−19.000622066253517166932315201310, −18.061411488866634647479708771743, −17.11541822782430112135641221227, −16.40278330722062080659517257132, −15.81087726094943643559204870410, −15.10148659775428453346014635752, −14.88742813571738560304416110677, −13.56636982308002101220479960685, −13.06969251317791984407397305689, −12.600347941705718455085577761948, −11.95851007739061685574196818316, −11.45906965112886288548217338783, −10.32637658668464219766804850833, −9.806812091005159845077467412142, −9.17396393263493425443733918585, −8.01236768018126769058304113116, −7.4449166818102002398426765040, −6.617211993618054324317127215134, −5.60162156513704414439533970762, −5.32003318803435175199907337048, −4.36003805965008405786338741731, −3.648112163036283103574970455042, −3.033308843610918988403709203961, −2.04349660321617359736276264455, −1.20254801602279840337084769757, 0.06718525665618533113382776219, 1.63562361582502610567484810485, 2.8546190034236900581846430900, 3.19539114026715289665205207937, 3.87039950189333433345603236873, 4.728346818717466961102960803044, 5.53824029103885000875337873028, 6.575871036780057522744060415006, 6.836419374302629977287655665476, 7.36856462018809370163095613088, 8.499096162806163914123320644058, 9.04399174702499817606011640212, 10.38637414885195260360975452193, 10.71392641028442732663543050356, 11.626733413848484585450202340765, 12.01841864594891833262272767686, 13.00544507520342388849630226692, 13.60317764359757643313663932074, 14.12648571759192562581705855352, 14.84181879984891232875551501640, 15.51007450009079625167885784658, 16.24195360116497292231352657038, 16.55287938204516335361126348462, 17.35738744899883236534574498667, 18.519757420647208940216281069699

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