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Properties

Label 1-4033-4033.2933-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.775 + 0.631i$
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  + (−0.642 + 0.766i)2-s + (−0.686 + 0.727i)3-s + (−0.173 − 0.984i)4-s + (0.893 − 0.448i)5-s + (−0.116 − 0.993i)6-s + (−0.835 − 0.549i)7-s + (0.866 + 0.5i)8-s + (−0.0581 − 0.998i)9-s + (−0.230 + 0.973i)10-s + (−0.230 − 0.973i)11-s + (0.835 + 0.549i)12-s + (−0.549 + 0.835i)13-s + (0.957 − 0.286i)14-s + (−0.286 + 0.957i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.686 + 0.727i)3-s + (−0.173 − 0.984i)4-s + (0.893 − 0.448i)5-s + (−0.116 − 0.993i)6-s + (−0.835 − 0.549i)7-s + (0.866 + 0.5i)8-s + (−0.0581 − 0.998i)9-s + (−0.230 + 0.973i)10-s + (−0.230 − 0.973i)11-s + (0.835 + 0.549i)12-s + (−0.549 + 0.835i)13-s + (0.957 − 0.286i)14-s + (−0.286 + 0.957i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2933, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.775 + 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2079086766 + 0.5843242760i\)
\(L(\frac12)\) \(\approx\) \(0.2079086766 + 0.5843242760i\)
\(L(1)\) \(\approx\) \(0.5729356629 + 0.2068909492i\)
\(L(1)\) \(\approx\) \(0.5729356629 + 0.2068909492i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (-0.686 + 0.727i)T \)
5 \( 1 + (0.893 - 0.448i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (-0.230 - 0.973i)T \)
13 \( 1 + (-0.549 + 0.835i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.686 - 0.727i)T \)
31 \( 1 + (0.0581 + 0.998i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.918 + 0.396i)T \)
53 \( 1 + (-0.998 - 0.0581i)T \)
59 \( 1 + (0.957 + 0.286i)T \)
61 \( 1 + (-0.597 - 0.802i)T \)
67 \( 1 + (0.448 - 0.893i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.973 - 0.230i)T \)
79 \( 1 + (0.549 + 0.835i)T \)
83 \( 1 + (0.686 - 0.727i)T \)
89 \( 1 + (0.597 + 0.802i)T \)
97 \( 1 + (0.893 - 0.448i)T \)
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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

−17.85781907368715191238550993981, −17.67924989536253801282997691623, −17.069706115727451300531814260445, −16.26207857519797629070179131091, −15.41429889745140045294003339897, −14.603601155110049191263431366502, −13.46225727643762339746559210052, −12.98357959954737032893103353541, −12.592062092807351620988071724789, −11.9585128216943522498874354167, −10.94414762117891876770615039530, −10.50523795434417782491854272630, −9.86230766885257165115427054036, −9.20011439555322326612663299770, −8.364157961435317531809535115982, −7.415363884679543742135782654771, −6.86597552999043643901266062004, −6.224485646030707899258226745863, −5.3000033083698235151641671201, −4.60286839754517559464370321392, −3.25999641403574388522639312832, −2.45300912634455914659038582272, −2.178468857184917843746563785767, −1.08569655368747394308476950795, −0.192776978722525037538282367894, 0.67997336374259336388600793758, 1.35559436165151939823218022906, 2.673761153075255464058183813814, 3.67124099712648486836789609481, 4.764123678253700399445544808, 5.12917280738831427105612834276, 6.0879807361546675209564928535, 6.41231307482330078662889859963, 7.19836975819240332354018515984, 8.20658037460537199435650878528, 9.18874669455643215823292836756, 9.48285837367879445689712029236, 10.04181560482936890248190948503, 10.75582155610249382838492392640, 11.455113074528858217674170064176, 12.37828069480002812273777428932, 13.34364896595320366740435739204, 13.96194825643444737321917972428, 14.43216819097444203524611577029, 15.56385135825388398252883263125, 16.21704316851798008777810904128, 16.3939059854446092129975200551, 17.115657868606873901694120947566, 17.60475951340300381496020880856, 18.35891744161627058350320577774

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