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Properties

Label 1-4033-4033.1506-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.114 - 0.993i$
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 + (0.597 + 0.802i)3-s + (0.173 + 0.984i)4-s + (−0.686 + 0.727i)5-s + (−0.0581 + 0.998i)6-s + (0.973 + 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.286 + 0.957i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (−0.686 + 0.727i)12-s + (0.973 + 0.230i)13-s + (0.597 + 0.802i)14-s + (−0.993 − 0.116i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.597 + 0.802i)3-s + (0.173 + 0.984i)4-s + (−0.686 + 0.727i)5-s + (−0.0581 + 0.998i)6-s + (0.973 + 0.230i)7-s + (−0.5 + 0.866i)8-s + (−0.286 + 0.957i)9-s + (−0.993 + 0.116i)10-s + (−0.993 − 0.116i)11-s + (−0.686 + 0.727i)12-s + (0.973 + 0.230i)13-s + (0.597 + 0.802i)14-s + (−0.993 − 0.116i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.114 - 0.993i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.114 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.251527495 + 1.403511338i\)
\(L(\frac12)\) \(\approx\) \(-1.251527495 + 1.403511338i\)
\(L(1)\) \(\approx\) \(0.7914832821 + 1.322523173i\)
\(L(1)\) \(\approx\) \(0.7914832821 + 1.322523173i\)

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 \( 1 + (0.597 + 0.802i)T \)
5 \( 1 + (-0.686 + 0.727i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.993 - 0.116i)T \)
13 \( 1 + (0.973 + 0.230i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.396 - 0.918i)T \)
31 \( 1 + (-0.686 - 0.727i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.835 + 0.549i)T \)
53 \( 1 + (0.973 + 0.230i)T \)
59 \( 1 + (-0.993 - 0.116i)T \)
61 \( 1 + (-0.0581 + 0.998i)T \)
67 \( 1 + (-0.286 + 0.957i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.993 - 0.116i)T \)
79 \( 1 + (-0.686 - 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (-0.835 - 0.549i)T \)
97 \( 1 + (0.973 + 0.230i)T \)
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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

−18.11743453571224544280413044485, −17.73885872877635593227247671182, −16.3616705275622266490402076480, −15.574086887987832855921711964936, −15.323108360478602546787463499411, −14.23755597730611162253649262714, −13.78654381151415453193158387083, −13.14623514420620056560370464806, −12.631415858859714144755144799041, −11.85893297956938652431739532420, −11.282849312966527222911651665760, −10.72578853557721899712835663789, −9.63567485174891795269167688511, −8.74210000631425600701665190506, −8.306221234088688761055086651946, −7.425753515756297529822812608791, −6.85547898857298219289646864561, −5.691065320477529261441378241587, −5.08799601538666202354712445787, −4.35644036295841777487641130849, −3.542141997021514638249829190575, −2.85640973746387464355755747835, −1.83507915203236214145544748013, −1.32642416018429342740471757740, −0.331333404386457577133413737410, 1.91887642270915820638529199164, 2.61615400046168207716557633341, 3.384211201685799890663399593992, 4.17415648065418032770520168303, 4.51117849888857731536152474231, 5.56341355844505280417429719163, 6.091858369255509793982841272043, 7.22266007806287090073106336063, 7.944175103483181175154587348, 8.2385623107721211609865072799, 8.94513091810087226096435513093, 10.16051796927459128736960393946, 10.83089686517897125658118529387, 11.44003192420785831151534866160, 11.97877977805215561241803936610, 13.185295492829423146368408991071, 13.70583762384808125198440830765, 14.35391255551361712849948863489, 14.945196428787453382076338378804, 15.506082197954112222569664320244, 15.93761860907994698954237348943, 16.53960856824739265360922971923, 17.58658959967660479782900603503, 18.29974563977599238534830760001, 18.755803872633268258570553187603

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