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Properties

Label 1-4024-4024.1275-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.955 + 0.294i$
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  + (−0.0812 + 0.996i)3-s + (−0.131 + 0.991i)5-s + (0.301 − 0.953i)7-s + (−0.986 − 0.161i)9-s + (−0.722 + 0.691i)11-s + (0.831 + 0.554i)13-s + (−0.977 − 0.211i)15-s + (−0.968 − 0.247i)17-s + (−0.265 − 0.964i)19-s + (0.925 + 0.378i)21-s + (−0.539 − 0.842i)23-s + (−0.965 − 0.259i)25-s + (0.241 − 0.970i)27-s + (0.962 − 0.271i)29-s + (−0.803 + 0.595i)31-s + ⋯
L(s)  = 1  + (−0.0812 + 0.996i)3-s + (−0.131 + 0.991i)5-s + (0.301 − 0.953i)7-s + (−0.986 − 0.161i)9-s + (−0.722 + 0.691i)11-s + (0.831 + 0.554i)13-s + (−0.977 − 0.211i)15-s + (−0.968 − 0.247i)17-s + (−0.265 − 0.964i)19-s + (0.925 + 0.378i)21-s + (−0.539 − 0.842i)23-s + (−0.965 − 0.259i)25-s + (0.241 − 0.970i)27-s + (0.962 − 0.271i)29-s + (−0.803 + 0.595i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.955 + 0.294i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.955 + 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103230975 + 0.1658756217i\)
\(L(\frac12)\) \(\approx\) \(1.103230975 + 0.1658756217i\)
\(L(1)\) \(\approx\) \(0.8480832371 + 0.2975752529i\)
\(L(1)\) \(\approx\) \(0.8480832371 + 0.2975752529i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.0812 + 0.996i)T \)
5 \( 1 + (-0.131 + 0.991i)T \)
7 \( 1 + (0.301 - 0.953i)T \)
11 \( 1 + (-0.722 + 0.691i)T \)
13 \( 1 + (0.831 + 0.554i)T \)
17 \( 1 + (-0.968 - 0.247i)T \)
19 \( 1 + (-0.265 - 0.964i)T \)
23 \( 1 + (-0.539 - 0.842i)T \)
29 \( 1 + (0.962 - 0.271i)T \)
31 \( 1 + (-0.803 + 0.595i)T \)
37 \( 1 + (-0.998 - 0.0625i)T \)
41 \( 1 + (0.507 + 0.861i)T \)
43 \( 1 + (0.731 - 0.682i)T \)
47 \( 1 + (0.610 + 0.791i)T \)
53 \( 1 + (0.265 - 0.964i)T \)
59 \( 1 + (0.168 + 0.985i)T \)
61 \( 1 + (0.630 - 0.776i)T \)
67 \( 1 + (-0.977 + 0.211i)T \)
71 \( 1 + (-0.372 - 0.928i)T \)
73 \( 1 + (0.713 - 0.700i)T \)
79 \( 1 + (-0.980 + 0.198i)T \)
83 \( 1 + (-0.205 + 0.978i)T \)
89 \( 1 + (0.845 - 0.533i)T \)
97 \( 1 + (0.992 - 0.124i)T \)
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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

−18.50456539432541686347990701557, −17.71130940998327261333687849804, −17.372529494736955627497658875048, −16.26908340944650755444318285754, −15.85728310849582922444918679061, −15.15693689336788091615553348166, −14.12136982631762753808585619927, −13.533384722917321959011008544481, −12.85899721128280921749236668491, −12.43594308793613557641444167976, −11.68428459739742914843507797766, −11.09370930322949370271827120564, −10.25803966611576972077650808110, −8.95749755720354840541686961282, −8.67658691981106885789741009960, −8.066938183973954515818492375851, −7.455189457867634292987416073966, −6.24285487251694667067039038078, −5.692103876081813539446153473100, −5.33819322305547584934666285273, −4.178730423584629707465752854, −3.26092319096622960188110653637, −2.27476101969677903645491529551, −1.66291720483558924699141754015, −0.74376913494415271194980592762, 0.41877547007119592686594421297, 1.9685356893642029873292424977, 2.70666346931185978088277607224, 3.55152877903176444068497031503, 4.3845085546463070102987445440, 4.65475354288304809744603917857, 5.80932864558192028790069267210, 6.68355294434371173848454116294, 7.129569270122024765624684184606, 8.115798227227796038503911434579, 8.82618329610355889520871461244, 9.661905754944458621470658442862, 10.49931245178272322270690770989, 10.744435908342404479549685929302, 11.30487499114805103983613782608, 12.15987624450782421509990272544, 13.2869479373700867526372525391, 13.88311319116648605074572807098, 14.46503679610904979715275975657, 15.15469946326028041445791464148, 15.867150449565116148928011431394, 16.18477266593185327255139474447, 17.272065171513634703249721157066, 17.80567219002115114086285968300, 18.24294932052689630753381513928

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