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Properties

Label 1-4033-4033.1367-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.999 - 0.0398i$
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  + (−0.5 + 0.866i)2-s + (−0.597 + 0.802i)3-s + (−0.5 − 0.866i)4-s + (0.230 + 0.973i)5-s + (−0.396 − 0.918i)6-s + (−0.286 + 0.957i)7-s + 8-s + (−0.286 − 0.957i)9-s + (−0.957 − 0.286i)10-s + (0.957 − 0.286i)11-s + (0.993 + 0.116i)12-s + (0.973 − 0.230i)13-s + (−0.686 − 0.727i)14-s + (−0.918 − 0.396i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.597 + 0.802i)3-s + (−0.5 − 0.866i)4-s + (0.230 + 0.973i)5-s + (−0.396 − 0.918i)6-s + (−0.286 + 0.957i)7-s + 8-s + (−0.286 − 0.957i)9-s + (−0.957 − 0.286i)10-s + (0.957 − 0.286i)11-s + (0.993 + 0.116i)12-s + (0.973 − 0.230i)13-s + (−0.686 − 0.727i)14-s + (−0.918 − 0.396i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.999 - 0.0398i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.999 - 0.0398i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02475554490 + 1.241789390i\)
\(L(\frac12)\) \(\approx\) \(0.02475554490 + 1.241789390i\)
\(L(1)\) \(\approx\) \(0.4893982045 + 0.6416919961i\)
\(L(1)\) \(\approx\) \(0.4893982045 + 0.6416919961i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (0.230 + 0.973i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (0.957 - 0.286i)T \)
13 \( 1 + (0.973 - 0.230i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.727 - 0.686i)T \)
31 \( 1 + (0.549 - 0.835i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.918 + 0.396i)T \)
53 \( 1 + (-0.918 + 0.396i)T \)
59 \( 1 + (0.396 - 0.918i)T \)
61 \( 1 + (-0.549 - 0.835i)T \)
67 \( 1 + (-0.116 - 0.993i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.686 - 0.727i)T \)
79 \( 1 + (0.597 + 0.802i)T \)
83 \( 1 + (0.835 - 0.549i)T \)
89 \( 1 + (0.957 + 0.286i)T \)
97 \( 1 + (-0.549 - 0.835i)T \)
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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

−18.01798091393276557874117731533, −17.62919011617950692459824819406, −16.81514629474191852402394710625, −16.47652121666211673877230520395, −15.917672382157814347210832817, −14.18142943398600841021046234917, −13.73533114185941207358427560036, −13.2871897381352549164746591771, −12.4221589855759084776745741507, −11.987939042938070542295798949851, −11.401888789366866000183076604843, −10.5268514751523472527854453832, −9.96439421222948216314589074414, −9.03944363640602541197485295023, −8.56452514028214962092429936366, −7.6803229410456534459598867586, −6.94492507340227415147974298657, −6.36388548045940140188697449174, −5.128137563265409832888875284728, −4.61241045638497408077504471669, −3.7441998673217355605585900603, −2.833130918547512681814833407842, −1.71646540569873675411012860459, −1.06365346125995270332051008426, −0.665758246754776993751413444449, 0.96222674729561064179823277892, 1.949104529226246684669289848610, 3.38161402126170337501405298182, 3.69554093519060816504009121884, 4.870570916757049124784322332354, 5.83477169932191659350801711315, 6.085338604237328558277180446166, 6.51803721084686800791356333468, 7.70294396225930704833170860757, 8.39414515085157261356856087099, 9.28251419273081966263313361155, 9.721917067021015931879562557921, 10.33912989966209042146308653974, 11.17340755739595826554618554131, 11.659706804059070896166509558984, 12.57680622081032097700588168007, 13.8147872820873370144450319945, 14.16413619831785204314166215437, 15.152266455966432057430985521250, 15.36863110817951238166422759712, 16.02045520657180789876011280815, 16.787439379391475685282581101190, 17.44005868982453375303635865857, 17.943719501520437521271932377056, 18.756218232460606749161884223471

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