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Properties

Label 1-4029-4029.437-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.220 - 0.975i$
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  + (0.584 − 0.811i)2-s + (−0.316 − 0.948i)4-s + (−0.508 − 0.860i)5-s + (−0.576 + 0.817i)7-s + (−0.954 − 0.297i)8-s + (−0.995 − 0.0904i)10-s + (0.0100 − 0.999i)11-s + (−0.979 − 0.200i)13-s + (0.326 + 0.945i)14-s + (−0.799 + 0.600i)16-s + (0.482 + 0.875i)19-s + (−0.655 + 0.755i)20-s + (−0.805 − 0.592i)22-s + (0.991 + 0.130i)23-s + (−0.482 + 0.875i)25-s + (−0.735 + 0.678i)26-s + ⋯
L(s)  = 1  + (0.584 − 0.811i)2-s + (−0.316 − 0.948i)4-s + (−0.508 − 0.860i)5-s + (−0.576 + 0.817i)7-s + (−0.954 − 0.297i)8-s + (−0.995 − 0.0904i)10-s + (0.0100 − 0.999i)11-s + (−0.979 − 0.200i)13-s + (0.326 + 0.945i)14-s + (−0.799 + 0.600i)16-s + (0.482 + 0.875i)19-s + (−0.655 + 0.755i)20-s + (−0.805 − 0.592i)22-s + (0.991 + 0.130i)23-s + (−0.482 + 0.875i)25-s + (−0.735 + 0.678i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.220 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9929601100 - 1.242682790i\)
\(L(\frac12)\) \(\approx\) \(0.9929601100 - 1.242682790i\)
\(L(1)\) \(\approx\) \(0.9235176025 - 0.6327825681i\)
\(L(1)\) \(\approx\) \(0.9235176025 - 0.6327825681i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.584 - 0.811i)T \)
5 \( 1 + (-0.508 - 0.860i)T \)
7 \( 1 + (-0.576 + 0.817i)T \)
11 \( 1 + (0.0100 - 0.999i)T \)
13 \( 1 + (-0.979 - 0.200i)T \)
19 \( 1 + (0.482 + 0.875i)T \)
23 \( 1 + (0.991 + 0.130i)T \)
29 \( 1 + (0.0503 + 0.998i)T \)
31 \( 1 + (0.957 - 0.287i)T \)
37 \( 1 + (0.931 - 0.364i)T \)
41 \( 1 + (0.268 + 0.963i)T \)
43 \( 1 + (0.927 + 0.373i)T \)
47 \( 1 + (-0.721 - 0.692i)T \)
53 \( 1 + (0.678 + 0.735i)T \)
59 \( 1 + (-0.446 - 0.894i)T \)
61 \( 1 + (0.805 - 0.592i)T \)
67 \( 1 + (0.354 + 0.935i)T \)
71 \( 1 + (0.0904 + 0.995i)T \)
73 \( 1 + (0.190 - 0.981i)T \)
83 \( 1 + (-0.446 + 0.894i)T \)
89 \( 1 + (-0.464 + 0.885i)T \)
97 \( 1 + (0.988 + 0.150i)T \)
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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

−18.59660177259056475555046747315, −17.63922874453250625407741085977, −17.337712032211255426086861658715, −16.57646172612855529358548363356, −15.70355448073191845465540882913, −15.32328090599616631317875235545, −14.61166904325550965137547348423, −14.05028690850064323523762256739, −13.30554965784731031194234833870, −12.657537233068120128379299761539, −11.90078512361348034329015871663, −11.29009045562135979980360234316, −10.28146065397187238817242794989, −9.69332708322650099048347846644, −8.892815452409265831086459744691, −7.64373828892579858139938757480, −7.466428316562241171832744686400, −6.770554090914639321728743476, −6.27088811091881053469207461799, −5.060354200501786702508581578967, −4.45641294914468242925487015555, −3.843611116723353736239985760274, −2.87766667961550534259362423894, −2.43216788289633299299257197829, −0.64194915268825554637691460757, 0.61034546219906908538615708090, 1.39539988528248919291284056554, 2.57068141715398549431077277935, 3.10897571210620931663106506302, 3.88585994963954958189120122533, 4.77939638434394301736509587222, 5.41328377343818730810253071762, 5.94269359927715704316257274504, 6.91367127199787084175051021593, 8.03348653240467867094517324815, 8.66041859807544779398998035479, 9.46731179665150752017562601717, 9.83967169286243147257728341187, 10.96152995240412194233926502312, 11.53605595712344426768974276926, 12.20853051937011892931564705200, 12.71022374949069965907023383601, 13.21818344224848674212470143053, 14.11722562006650114497446754397, 14.83077268263601777294246725547, 15.45811029959943232612776178913, 16.22347546839832619868329695438, 16.70496218409248306150377433610, 17.74816309068501678628108410743, 18.596907222265736579297005618583

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