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Properties

Label 1-4024-4024.445-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.792 + 0.610i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (0.349 + 0.937i)3-s + (0.590 − 0.806i)5-s + (0.580 + 0.814i)7-s + (−0.756 + 0.654i)9-s + (0.155 + 0.987i)11-s + (0.986 − 0.161i)13-s + (0.962 + 0.271i)15-s + (−0.229 + 0.973i)17-s + (−0.838 − 0.544i)19-s + (−0.560 + 0.828i)21-s + (0.600 + 0.799i)23-s + (−0.301 − 0.953i)25-s + (−0.877 − 0.479i)27-s + (−0.747 + 0.663i)29-s + (0.677 − 0.735i)31-s + ⋯
L(s)  = 1  + (0.349 + 0.937i)3-s + (0.590 − 0.806i)5-s + (0.580 + 0.814i)7-s + (−0.756 + 0.654i)9-s + (0.155 + 0.987i)11-s + (0.986 − 0.161i)13-s + (0.962 + 0.271i)15-s + (−0.229 + 0.973i)17-s + (−0.838 − 0.544i)19-s + (−0.560 + 0.828i)21-s + (0.600 + 0.799i)23-s + (−0.301 − 0.953i)25-s + (−0.877 − 0.479i)27-s + (−0.747 + 0.663i)29-s + (0.677 − 0.735i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.792 + 0.610i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.792 + 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6246691809 + 1.835034582i\)
\(L(\frac12)\) \(\approx\) \(0.6246691809 + 1.835034582i\)
\(L(1)\) \(\approx\) \(1.163363191 + 0.6002316255i\)
\(L(1)\) \(\approx\) \(1.163363191 + 0.6002316255i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.349 + 0.937i)T \)
5 \( 1 + (0.590 - 0.806i)T \)
7 \( 1 + (0.580 + 0.814i)T \)
11 \( 1 + (0.155 + 0.987i)T \)
13 \( 1 + (0.986 - 0.161i)T \)
17 \( 1 + (-0.229 + 0.973i)T \)
19 \( 1 + (-0.838 - 0.544i)T \)
23 \( 1 + (0.600 + 0.799i)T \)
29 \( 1 + (-0.747 + 0.663i)T \)
31 \( 1 + (0.677 - 0.735i)T \)
37 \( 1 + (-0.900 - 0.435i)T \)
41 \( 1 + (-0.772 + 0.635i)T \)
43 \( 1 + (-0.640 - 0.768i)T \)
47 \( 1 + (0.0187 + 0.999i)T \)
53 \( 1 + (-0.838 + 0.544i)T \)
59 \( 1 + (0.831 - 0.554i)T \)
61 \( 1 + (0.871 + 0.490i)T \)
67 \( 1 + (-0.962 + 0.271i)T \)
71 \( 1 + (-0.649 + 0.760i)T \)
73 \( 1 + (-0.372 + 0.928i)T \)
79 \( 1 + (-0.686 + 0.726i)T \)
83 \( 1 + (-0.517 + 0.855i)T \)
89 \( 1 + (-0.610 + 0.791i)T \)
97 \( 1 + (0.620 - 0.784i)T \)
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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

−18.29665324987893623723633515688, −17.63702812403791337734696623680, −17.01249701449631402580061830573, −16.35036180159289019290411680680, −15.277464924258045039353880037283, −14.48171316947998221435003966816, −14.10314508092931918923520945884, −13.378548259218178685044419054168, −13.18775045058458504785058178354, −11.837448690551827434051715745636, −11.39095199325837067650040664978, −10.67257625703562050178447349969, −10.06475181654310334006123699930, −8.85113542105546047597422140468, −8.525022845912153107570828293641, −7.640287383656395855440674349095, −6.86620416349350586327174538979, −6.43870776582475739312214519254, −5.72794671439845454387776610107, −4.7005585370111538672943644032, −3.54362550894294655421814284941, −3.13794260917422151605945203745, −2.05882162335636280269960901048, −1.477380052738956245909852531973, −0.46808675667026272873078938200, 1.44373367280386120628512512870, 1.9633857761898324053155413788, 2.86282486358742734651167903077, 3.95214472139401514444045573182, 4.49190916287275989477810352398, 5.292819534052180230367587961150, 5.74846481603850941153771284058, 6.6880750746025339697640130867, 7.88470479541805074679406109321, 8.65606845540302150492940692111, 8.86934375578050541987421671152, 9.678724307628436846177654105506, 10.35370239980434883854455218741, 11.14262057518090975188148304049, 11.75949238250074709429929710962, 12.84195779755731897546381581733, 13.11441032359244034205682532204, 14.11953364469491791365470313862, 14.78337626841846879983820147900, 15.46636028279816293702665069387, 15.732071630686652783986360979192, 16.85512741375831791445754075847, 17.294397141764554101818244482341, 17.8158307106575909987579525328, 18.788730974888733241349696729784

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