Properties

Label 1-4033-4033.783-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.364 + 0.931i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.984 + 0.173i)2-s + (0.993 − 0.116i)3-s + (0.939 + 0.342i)4-s + (0.957 − 0.286i)5-s + (0.998 + 0.0581i)6-s + (−0.686 + 0.727i)7-s + (0.866 + 0.5i)8-s + (0.973 − 0.230i)9-s + (0.993 − 0.116i)10-s + (−0.993 − 0.116i)11-s + (0.973 + 0.230i)12-s + (0.727 + 0.686i)13-s + (−0.802 + 0.597i)14-s + (0.918 − 0.396i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.993 − 0.116i)3-s + (0.939 + 0.342i)4-s + (0.957 − 0.286i)5-s + (0.998 + 0.0581i)6-s + (−0.686 + 0.727i)7-s + (0.866 + 0.5i)8-s + (0.973 − 0.230i)9-s + (0.993 − 0.116i)10-s + (−0.993 − 0.116i)11-s + (0.973 + 0.230i)12-s + (0.727 + 0.686i)13-s + (−0.802 + 0.597i)14-s + (0.918 − 0.396i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.364 + 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.702286689 + 5.258522199i\)
\(L(\frac12)\) \(\approx\) \(7.702286689 + 5.258522199i\)
\(L(1)\) \(\approx\) \(3.138273847 + 0.7788946417i\)
\(L(1)\) \(\approx\) \(3.138273847 + 0.7788946417i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (0.993 - 0.116i)T \)
5 \( 1 + (0.957 - 0.286i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (-0.993 - 0.116i)T \)
13 \( 1 + (0.727 + 0.686i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (-0.918 - 0.396i)T \)
31 \( 1 + (-0.727 + 0.686i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.835 - 0.549i)T \)
53 \( 1 + (0.286 + 0.957i)T \)
59 \( 1 + (0.918 - 0.396i)T \)
61 \( 1 + (-0.549 + 0.835i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.993 + 0.116i)T \)
79 \( 1 + (-0.230 - 0.973i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (0.998 + 0.0581i)T \)
97 \( 1 + (-0.230 + 0.973i)T \)
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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

−18.44110282587706874452566936954, −17.476297245750689217025054741751, −16.453668275536981316241025583010, −16.06884569526916034228739026137, −15.10349524521610231025243310355, −14.78663047343359251277146874339, −13.83113175214590294363705914335, −13.497499294862551171779506534483, −12.87755008963644824774541162320, −12.56437903880292601126106678741, −10.9479650966337832186534904183, −10.66340547152185208674666857115, −9.974040486266507841692046844751, −9.41018221620316403509733327992, −8.30111202848191299488131254482, −7.50697538061780238393434755326, −6.91909330223809952511633285912, −6.050813926351253947447271208091, −5.44417276596172730186166460249, −4.506987297581358129279786422774, −3.66397338310787678137174929038, −3.085004119364948574612667877, −2.46679269985950479875661436036, −1.66288615278986911503033193949, −0.69010265351603144025559879245, 1.18227843284044055515826035975, 2.04181974617754445400097404050, 2.53562561130218194792703953866, 3.40036884149346202783516665216, 3.91691716332910233476023876784, 5.18347868219268891825900498685, 5.60011685927710954473934680783, 6.29335217773366848710271141077, 7.214651733894043844513039528578, 7.805957566880756359141146583951, 8.753816677693032232692018492073, 9.346443173246870847968810117862, 10.0633442627487696964247703158, 10.80919273829775776209320086175, 12.02138676130615075613871381833, 12.39773723061168063459441707488, 13.32089294765817406174943419459, 13.54065315154656609864150359923, 14.18119444925753385903944466191, 14.87852685079074117645249443009, 15.74505943539740722313860069400, 16.094316242796894880837519782265, 16.72645913105690989732532537896, 17.88750975186299960752731198435, 18.62634512456941380065012698673

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