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Properties

Label 1-4033-4033.569-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.989 + 0.142i$
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

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.989 + 0.142i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.989 + 0.142i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.850908980 + 0.1324549518i\)
\(L(\frac12)\) \(\approx\) \(1.850908980 + 0.1324549518i\)
\(L(1)\) \(\approx\) \(1.097362239 + 0.02623520983i\)
\(L(1)\) \(\approx\) \(1.097362239 + 0.02623520983i\)

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 \( 1 + (0.286 + 0.957i)T \)
5 \( 1 + (0.998 + 0.0581i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (0.727 - 0.686i)T \)
13 \( 1 + (-0.893 + 0.448i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.957 + 0.286i)T \)
31 \( 1 + (0.549 + 0.835i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.802 + 0.597i)T \)
53 \( 1 + (0.549 - 0.835i)T \)
59 \( 1 + (-0.973 + 0.230i)T \)
61 \( 1 + (0.116 + 0.993i)T \)
67 \( 1 + (0.998 - 0.0581i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.686 + 0.727i)T \)
79 \( 1 + (-0.893 - 0.448i)T \)
83 \( 1 + (0.286 + 0.957i)T \)
89 \( 1 + (-0.116 - 0.993i)T \)
97 \( 1 + (0.998 + 0.0581i)T \)
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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

−18.321441515361917822285450466878, −17.77589509159183918072576368140, −17.12769209419943626234441359413, −17.00181520273881024671777814078, −15.605200225033549178489828750894, −14.8519095420472347875844825892, −14.601778788148100706203223319813, −13.73875871247742006960279976674, −13.243395352393146453241885990756, −12.22281305503612592065814678144, −11.567793629359793685987056944611, −10.89254154333178659321061688142, −9.66826268063297881391659851940, −9.44136179856298781021277535835, −8.73214434296415099104682932722, −7.82750740152301203334600302879, −7.38679314867586128004179090167, −6.637633872479203347675843143034, −5.92084942284549286235830321265, −5.26069595511342129432049294016, −4.57866621723552135958823144072, −2.92331892978991484135458273270, −2.1258640799738995135296733769, −1.66294160643535839100209343222, −0.8449656269319145602837220462, 0.82768247248852148985024170649, 1.81438754242428333848270464227, 2.45579432304696112737810150723, 3.283591423874436615945997356520, 4.1484767328301519121481984987, 4.81488595803987564919073951402, 5.59562308642765096091975506899, 6.79292259320480368386382757063, 7.33805616750068456160417000886, 8.54372039593554065818200906287, 8.85545521451783054140090421789, 9.46357797860668087800777314015, 10.23391517734389418699377712322, 10.774375615476809083655906838449, 11.32727257855756218249680989942, 12.014782686844096776696756583289, 13.087363685963950771813571101752, 13.86251827599179203593080703138, 14.25445176152948803605314800997, 15.06064589701158923592770237648, 15.96945399082949260938931782382, 16.79509273492429074097554681242, 17.15907916231552852495064501690, 17.586616805779421589309399056646, 18.49991590271893890755138833602

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