↑

Properties

Label 1-4033-4033.1034-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.350 - 0.936i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.396 + 0.918i)3-s + (0.173 − 0.984i)4-s + (−0.727 − 0.686i)5-s + (−0.286 − 0.957i)6-s + (0.973 + 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (0.835 + 0.549i)12-s + (0.286 + 0.957i)13-s + (−0.893 + 0.448i)14-s + (0.918 − 0.396i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.396 + 0.918i)3-s + (0.173 − 0.984i)4-s + (−0.727 − 0.686i)5-s + (−0.286 − 0.957i)6-s + (0.973 + 0.230i)7-s + (0.5 + 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.998 + 0.0581i)10-s + (−0.998 + 0.0581i)11-s + (0.835 + 0.549i)12-s + (0.286 + 0.957i)13-s + (−0.893 + 0.448i)14-s + (0.918 − 0.396i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1034, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006490229774 + 0.009355988602i\)
\(L(\frac12)\) \(\approx\) \(0.006490229774 + 0.009355988602i\)
\(L(1)\) \(\approx\) \(0.4747382687 + 0.2162616317i\)
\(L(1)\) \(\approx\) \(0.4747382687 + 0.2162616317i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (-0.396 + 0.918i)T \)
5 \( 1 + (-0.727 - 0.686i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.998 + 0.0581i)T \)
13 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 - T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.998 - 0.0581i)T \)
31 \( 1 + (0.802 + 0.597i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.727 + 0.686i)T \)
53 \( 1 + (-0.549 + 0.835i)T \)
59 \( 1 + (-0.396 - 0.918i)T \)
61 \( 1 + (0.549 - 0.835i)T \)
67 \( 1 + (-0.998 + 0.0581i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.835 - 0.549i)T \)
79 \( 1 + (-0.893 - 0.448i)T \)
83 \( 1 + (0.286 + 0.957i)T \)
89 \( 1 + (-0.918 - 0.396i)T \)
97 \( 1 + (-0.116 - 0.993i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.589877622956955644282866901153, −18.19153549986067128813297670544, −17.4747932026723975090278301335, −17.01683747453404025269604026157, −16.16534239213867671698111487977, −15.21027564887124940200621823696, −14.75057598868149343358512131509, −13.57917442735903331835477441661, −13.03687180010519869825301975711, −12.41980372643313183256170948473, −11.557828297540470040497502641453, −11.16971014144230884444210118703, −10.55989186899193204223497713532, −10.069920527610432342655084057349, −8.52102741735762285894871362950, −8.23723236608797769172442736484, −7.58030067746777743297930088554, −7.18848375505942013780187962627, −6.138689632546610837583629292795, −5.292124432303507805902940169557, −4.29737169164935117446794020236, −3.331905636255361072396519515474, −2.67123318950759299064475209014, −1.82741116552764687397084272777, −1.007942436376918164302338447662, 0.00557501012109602161322477916, 1.07926886947427282709384929591, 2.0698552068209793073577110260, 3.2325423814312432052879131997, 4.48120153175277326021912273842, 4.76243425375171775265599210960, 5.402388114241376868496626846600, 6.23808959432398872631057473360, 7.21018064157947360427174755950, 7.9563379614402017310085536819, 8.59889436641246786206235605650, 9.04677446205649610084349302267, 9.85994794987524084391227301723, 10.742458583030199554150807171034, 11.14391567203128489325650612814, 11.81987353295057543425978967760, 12.59700989145094993534465508519, 13.77416847451680983817533825303, 14.52622214590033126490464720570, 15.120351172073672099730615111953, 15.685537438923684195854931937194, 16.22433593991570149869825413846, 16.94944725920636071009984721686, 17.21954448356387484595875278939, 18.27573620774738658192247409858

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