↑

Properties

Label 1-4033-4033.1565-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.996 + 0.0834i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 − 0.766i)2-s + (0.973 − 0.230i)3-s + (−0.173 + 0.984i)4-s + (0.835 − 0.549i)5-s + (−0.802 − 0.597i)6-s + (−0.0581 − 0.998i)7-s + (0.866 − 0.5i)8-s + (0.893 − 0.448i)9-s + (−0.957 − 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.0581 + 0.998i)12-s + (0.998 − 0.0581i)13-s + (−0.727 + 0.686i)14-s + (0.686 − 0.727i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (0.973 − 0.230i)3-s + (−0.173 + 0.984i)4-s + (0.835 − 0.549i)5-s + (−0.802 − 0.597i)6-s + (−0.0581 − 0.998i)7-s + (0.866 − 0.5i)8-s + (0.893 − 0.448i)9-s + (−0.957 − 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.0581 + 0.998i)12-s + (0.998 − 0.0581i)13-s + (−0.727 + 0.686i)14-s + (0.686 − 0.727i)15-s + (−0.939 − 0.342i)16-s + (−0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.996 + 0.0834i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.996 + 0.0834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1167422870 - 2.792115078i\)
\(L(\frac12)\) \(\approx\) \(-0.1167422870 - 2.792115078i\)
\(L(1)\) \(\approx\) \(0.9735207717 - 0.8453803866i\)
\(L(1)\) \(\approx\) \(0.9735207717 - 0.8453803866i\)

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 \( 1 + (0.973 - 0.230i)T \)
5 \( 1 + (0.835 - 0.549i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (-0.957 + 0.286i)T \)
13 \( 1 + (0.998 - 0.0581i)T \)
17 \( 1 + (-0.342 + 0.939i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (0.973 - 0.230i)T \)
31 \( 1 + (0.893 - 0.448i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.116 - 0.993i)T \)
53 \( 1 + (-0.448 + 0.893i)T \)
59 \( 1 + (-0.727 - 0.686i)T \)
61 \( 1 + (0.396 + 0.918i)T \)
67 \( 1 + (-0.549 + 0.835i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.286 - 0.957i)T \)
79 \( 1 + (-0.998 - 0.0581i)T \)
83 \( 1 + (-0.973 + 0.230i)T \)
89 \( 1 + (-0.396 - 0.918i)T \)
97 \( 1 + (0.835 - 0.549i)T \)
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

−18.61883627364463300091546396857, −18.12304291043988774556100956744, −17.48614323492911912683985229578, −16.35571965455601608340853473786, −15.68045691706193608122654245718, −15.57542920766465316934751228675, −14.557528192643340127914399148258, −14.04763411856662129969661231426, −13.503062804639679880236764410931, −12.84791593127894282261893944517, −11.554733465384265191268898137728, −10.75858299403126779139809488535, −10.07660994222705848022587337334, −9.55452938080821207144637792829, −8.81870667616425318261767409606, −8.36964324304165907942285095354, −7.58107374566141514348702199667, −6.78893073895212405920495390569, −6.0506099768783815374364030785, −5.35317460160703045081708095990, −4.71421434094049052970925228974, −3.28331688046325406062011840498, −2.769967582767478475423551211708, −1.89110584371618613034202521115, −1.16298633745277134551764962056, 0.42896244352071281188167356958, 1.160922779030637843772661792420, 1.7996230201324680616108406588, 2.71392079586147934445676399733, 3.23709095751899490824152071727, 4.389076339367802378675030098700, 4.672358178158872710230876342472, 6.19452520697141370740310416439, 6.90981196800449998902631227686, 7.71237896935350624802862627558, 8.509110947857966703924204375, 8.74001817601730078933875094014, 9.75802199105490940541545688867, 10.25511027366615746866890251468, 10.71814717678874824274885204414, 11.75272149403527419025901567131, 12.7874179018813857363164672733, 13.25008468781760872306523270821, 13.43663599454982565144423041100, 14.25047919498001869275143379048, 15.35675110262870780349545265282, 15.983262807738790980903332963200, 16.817536597204048275855946826356, 17.44306903486091614670201455137, 18.08208344845375468820714754532

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