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Properties

Label 1-4033-4033.3063-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.452 + 0.891i$
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 + (0.686 + 0.727i)3-s + (0.173 − 0.984i)4-s + (0.448 − 0.893i)5-s + (−0.993 − 0.116i)6-s + (−0.835 + 0.549i)7-s + (0.5 + 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.230 + 0.973i)10-s + (−0.230 + 0.973i)11-s + (0.835 − 0.549i)12-s + (0.835 − 0.549i)13-s + (0.286 − 0.957i)14-s + (0.957 − 0.286i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.686 + 0.727i)3-s + (0.173 − 0.984i)4-s + (0.448 − 0.893i)5-s + (−0.993 − 0.116i)6-s + (−0.835 + 0.549i)7-s + (0.5 + 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.230 + 0.973i)10-s + (−0.230 + 0.973i)11-s + (0.835 − 0.549i)12-s + (0.835 − 0.549i)13-s + (0.286 − 0.957i)14-s + (0.957 − 0.286i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.452 + 0.891i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.452 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7530089236 + 1.226747085i\)
\(L(\frac12)\) \(\approx\) \(0.7530089236 + 1.226747085i\)
\(L(1)\) \(\approx\) \(0.8283246276 + 0.4725936274i\)
\(L(1)\) \(\approx\) \(0.8283246276 + 0.4725936274i\)

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 \( 1 + (0.686 + 0.727i)T \)
5 \( 1 + (0.448 - 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.230 + 0.973i)T \)
13 \( 1 + (0.835 - 0.549i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.727 + 0.686i)T \)
31 \( 1 + (0.998 + 0.0581i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.918 + 0.396i)T \)
53 \( 1 + (0.998 - 0.0581i)T \)
59 \( 1 + (0.286 + 0.957i)T \)
61 \( 1 + (-0.802 - 0.597i)T \)
67 \( 1 + (0.448 + 0.893i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.973 - 0.230i)T \)
79 \( 1 + (0.835 + 0.549i)T \)
83 \( 1 + (0.686 + 0.727i)T \)
89 \( 1 + (0.802 + 0.597i)T \)
97 \( 1 + (0.448 - 0.893i)T \)
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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

−18.55583068377042450471288702013, −17.7726262725206585199048266330, −17.1609837468337058246723995637, −16.30266102104165554563398760331, −15.65962705731841416006399905743, −14.77322495197468156138184387197, −13.67146453569866384703692217822, −13.508258981294402608847975218722, −13.02756302619928636619806739599, −11.92266671379056645162572716365, −11.22151312285333103659358955898, −10.71091921735775038740140400519, −9.85772331295081779677694690640, −9.222241746083861611373542463929, −8.6584216094556428361760064997, −7.80628246637151731726829485888, −7.118210288979653793147080318196, −6.47830772624019356506787093216, −6.0371978299531405978047740861, −4.28482328304467827959144842931, −3.48925060990455524451220732486, −2.95338271248054412097749499449, −2.37719022534055035224829311095, −1.37691277081240115387157289018, −0.56375179510991303847903161423, 0.948108116989403529231509683004, 1.96394788738949800638965867293, 2.647083329112714219171815369265, 3.709211086474491727721767201084, 4.68056125076829760950774274884, 5.390123899842540990669431656801, 5.835265371639873286094190615385, 6.96605326430951150858056503805, 7.60833185701223217114838536137, 8.545251700748948547723192814456, 9.051060632132599917112855324372, 9.408000721158796183815609787625, 10.1510814138801199992767452682, 10.69395956361791895425944304555, 11.767738109983303663282552064024, 12.69714894682130786658209344695, 13.53301177956829531484658151690, 13.81325544550465973459965569642, 15.07155046647054888008545345850, 15.39786015765183168014138652942, 15.98323700462152392993009947807, 16.44050085282638504215327289306, 17.31477605398430697251656587318, 17.89071946294856585295675511144, 18.64315546292575957720712625455

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