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Properties

Label 1-4029-4029.2444-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.329 + 0.944i$
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  + (0.428 + 0.903i)2-s + (−0.632 + 0.774i)4-s + (0.979 + 0.200i)5-s + (−0.721 + 0.692i)7-s + (−0.970 − 0.239i)8-s + (0.239 + 0.970i)10-s + (−0.316 + 0.948i)11-s + (0.987 + 0.160i)13-s + (−0.935 − 0.354i)14-s + (−0.200 − 0.979i)16-s + (0.919 − 0.391i)19-s + (−0.774 + 0.632i)20-s + (−0.992 + 0.120i)22-s + (0.866 − 0.5i)23-s + (0.919 + 0.391i)25-s + (0.278 + 0.960i)26-s + ⋯
L(s)  = 1  + (0.428 + 0.903i)2-s + (−0.632 + 0.774i)4-s + (0.979 + 0.200i)5-s + (−0.721 + 0.692i)7-s + (−0.970 − 0.239i)8-s + (0.239 + 0.970i)10-s + (−0.316 + 0.948i)11-s + (0.987 + 0.160i)13-s + (−0.935 − 0.354i)14-s + (−0.200 − 0.979i)16-s + (0.919 − 0.391i)19-s + (−0.774 + 0.632i)20-s + (−0.992 + 0.120i)22-s + (0.866 − 0.5i)23-s + (0.919 + 0.391i)25-s + (0.278 + 0.960i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.329 + 0.944i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (2444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.329 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.529373077 + 2.154313078i\)
\(L(\frac12)\) \(\approx\) \(1.529373077 + 2.154313078i\)
\(L(1)\) \(\approx\) \(1.192868141 + 0.9201138888i\)
\(L(1)\) \(\approx\) \(1.192868141 + 0.9201138888i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.428 + 0.903i)T \)
5 \( 1 + (0.979 + 0.200i)T \)
7 \( 1 + (-0.721 + 0.692i)T \)
11 \( 1 + (-0.316 + 0.948i)T \)
13 \( 1 + (0.987 + 0.160i)T \)
19 \( 1 + (0.919 - 0.391i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.999 - 0.0402i)T \)
31 \( 1 + (0.0804 - 0.996i)T \)
37 \( 1 + (0.600 - 0.799i)T \)
41 \( 1 + (-0.663 + 0.748i)T \)
43 \( 1 + (0.948 - 0.316i)T \)
47 \( 1 + (0.799 - 0.600i)T \)
53 \( 1 + (-0.278 - 0.960i)T \)
59 \( 1 + (0.632 + 0.774i)T \)
61 \( 1 + (0.992 + 0.120i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (0.239 - 0.970i)T \)
73 \( 1 + (0.160 + 0.987i)T \)
83 \( 1 + (-0.632 + 0.774i)T \)
89 \( 1 + (0.970 - 0.239i)T \)
97 \( 1 + (0.992 + 0.120i)T \)
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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

−18.59181150855093516801795530422, −17.592879427767620103879918170657, −17.12251708578483917874490285176, −16.019832458544461374927311989798, −15.77048406140491157942226838834, −14.410964254709734449728849999256, −13.93646880490204026803488288793, −13.42509608670971000249905807284, −12.97982671255424662522159874254, −12.195344846714934355680642975906, −11.28302602859023357264619238787, −10.588292605833921963766656940020, −10.21949053450546664628145519123, −9.33891628788413488797460062809, −8.87137572645704536386866650498, −7.9427054635419527450101338473, −6.70683726663195862693440249549, −6.13120991526099383836486556947, −5.456684674606809623099387392668, −4.77676533396615564515329209843, −3.66650952815345546865702268776, −3.19923000859271790791746004675, −2.47497903447494182585029287476, −1.12844165293951395148651419200, −0.99624894407750184349913891747, 0.88977141259649427750128978413, 2.29179818950426271792571980322, 2.80318320935835391965950825239, 3.73149087020626445693430394903, 4.67501852661585571699035505296, 5.44980443098950246858127155890, 5.93790232244365746002234031828, 6.7432449118090814205631097644, 7.13946850655296695574182165663, 8.260340736476141059989679629847, 8.9382866552850951858708548408, 9.56249621802532273679432856155, 10.1042701122938330333074103916, 11.20342080003769946869662201241, 12.06387238592905839764825164613, 12.87667990542634949305916556286, 13.2079147189826463229480545157, 13.906481498582636248662461693, 14.67530001490042937619097284507, 15.28708418281901141754394110692, 15.925690822238751520650053955259, 16.493882247636081065183373052891, 17.327203237953400913826406911227, 17.944377580061299884481262760323, 18.40536399611041613618550027280

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