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Properties

Label 1-4024-4024.1155-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.599 + 0.800i$
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  + (−0.915 − 0.401i)3-s + (0.910 + 0.412i)5-s + (−0.452 + 0.891i)7-s + (0.677 + 0.735i)9-s + (0.560 − 0.828i)11-s + (0.803 − 0.595i)13-s + (−0.668 − 0.743i)15-s + (0.998 + 0.0625i)17-s + (−0.947 − 0.319i)19-s + (0.772 − 0.635i)21-s + (−0.968 − 0.247i)23-s + (0.659 + 0.752i)25-s + (−0.325 − 0.945i)27-s + (0.0687 + 0.997i)29-s + (0.810 + 0.585i)31-s + ⋯
L(s)  = 1  + (−0.915 − 0.401i)3-s + (0.910 + 0.412i)5-s + (−0.452 + 0.891i)7-s + (0.677 + 0.735i)9-s + (0.560 − 0.828i)11-s + (0.803 − 0.595i)13-s + (−0.668 − 0.743i)15-s + (0.998 + 0.0625i)17-s + (−0.947 − 0.319i)19-s + (0.772 − 0.635i)21-s + (−0.968 − 0.247i)23-s + (0.659 + 0.752i)25-s + (−0.325 − 0.945i)27-s + (0.0687 + 0.997i)29-s + (0.810 + 0.585i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.599 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.326042853 + 0.6637492592i\)
\(L(\frac12)\) \(\approx\) \(1.326042853 + 0.6637492592i\)
\(L(1)\) \(\approx\) \(0.9802177847 + 0.09463087882i\)
\(L(1)\) \(\approx\) \(0.9802177847 + 0.09463087882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.915 - 0.401i)T \)
5 \( 1 + (0.910 + 0.412i)T \)
7 \( 1 + (-0.452 + 0.891i)T \)
11 \( 1 + (0.560 - 0.828i)T \)
13 \( 1 + (0.803 - 0.595i)T \)
17 \( 1 + (0.998 + 0.0625i)T \)
19 \( 1 + (-0.947 - 0.319i)T \)
23 \( 1 + (-0.968 - 0.247i)T \)
29 \( 1 + (0.0687 + 0.997i)T \)
31 \( 1 + (0.810 + 0.585i)T \)
37 \( 1 + (0.695 + 0.718i)T \)
41 \( 1 + (-0.864 + 0.501i)T \)
43 \( 1 + (-0.982 + 0.186i)T \)
47 \( 1 + (0.528 + 0.848i)T \)
53 \( 1 + (0.947 - 0.319i)T \)
59 \( 1 + (0.939 + 0.343i)T \)
61 \( 1 + (0.845 + 0.533i)T \)
67 \( 1 + (-0.668 + 0.743i)T \)
71 \( 1 + (-0.883 + 0.468i)T \)
73 \( 1 + (0.192 + 0.981i)T \)
79 \( 1 + (-0.998 + 0.0500i)T \)
83 \( 1 + (0.429 - 0.902i)T \)
89 \( 1 + (-0.600 + 0.799i)T \)
97 \( 1 + (-0.0312 - 0.999i)T \)
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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

−18.198562462823289754390194716813, −17.494470614107787993326449237085, −16.85828632156609680893865350216, −16.677001831901479052251973746285, −15.863522572689240368444701872392, −15.02180280256174075795299245729, −14.21470725390721246465664514891, −13.51007686876357027070364224714, −12.93607489117061938053756480672, −12.071662745176124688856149983200, −11.650595065238855361200580166433, −10.51685720532339722102931527595, −10.11144410753943004713096716301, −9.64861500065359895021721333929, −8.853604406708039408819037728345, −7.82749999084985950321515930414, −6.85865600934480436550276056398, −6.30483789949925154111940953417, −5.79966615654762280986227532853, −4.85257434104530829851717187605, −4.077268205546841844842418576426, −3.71061683176577732793060024663, −2.178030956531585524685238401086, −1.44083437687043763749324703792, −0.56106578726577305744894876417, 0.97099710818198749644283860096, 1.65908597313477584782356166857, 2.687258905808984943213999388141, 3.328155641741163850744569004241, 4.47899109990856368218640109205, 5.625142971809353605667531650744, 5.765076788533483280016219951121, 6.476551225654969969922592304604, 7.018112281250689197130492659315, 8.376736643379867335649025964298, 8.624227815077494477684674534456, 9.94678912762576472675205061588, 10.15977510395489660696255438643, 11.088426029584476388732475723841, 11.689509698284124755446132967, 12.42143026468240151827414730157, 13.09601121080380299516746516019, 13.61080128464691033484257185688, 14.47460578151304364161854661355, 15.16197908473398866095976417811, 16.15802823469458767431190755214, 16.48791820271144259296221916676, 17.309446765089708672187874720687, 17.96336427840387556133745835160, 18.495055264159348059349589382889

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