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Properties

Label 1-4033-4033.578-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.782 + 0.622i$
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 + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s − 6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + i·10-s + i·11-s + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + 14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s − 6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + i·10-s + i·11-s + (0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + 14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.782 + 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03891355051 + 0.1114597797i\)
\(L(\frac12)\) \(\approx\) \(0.03891355051 + 0.1114597797i\)
\(L(1)\) \(\approx\) \(0.5671553589 - 0.2084425543i\)
\(L(1)\) \(\approx\) \(0.5671553589 - 0.2084425543i\)

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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + iT \)
31 \( 1 + iT \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 - iT \)
47 \( 1 + iT \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + iT \)
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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

−18.1875978314231846256159435885, −17.304132409532475822464849599350, −16.59661173166725253416815487602, −16.1690976990412732586391490031, −15.5372957382794802877507938957, −15.0122803883391013242964459762, −14.30306088294714613414143607478, −13.69532340447124240532162004152, −13.1215187814055585976543054930, −11.740529924538374110809959227864, −11.052965321360640732733502535245, −10.337215653895555198103414107194, −9.96836080632111824783673688773, −9.09202525961188753478719333124, −8.380201771218345806403188858730, −7.573217420001537099615839246905, −7.41222910850365811078066084394, −6.21338050705822077792620931786, −5.61351751542371933528672800061, −4.48830250014187242895710959643, −4.115582706204354341676124699329, −3.16611678709669767122232137916, −2.51005355500843571207074558378, −0.74269639482983594559759486104, −0.051553460794326067104246056326, 1.4404717833947381166864991844, 1.847675247280926045619353656598, 2.72580976387412333892950097793, 3.58119723231318419659700764729, 4.16823361537811471118078002781, 5.13300961392030048529287858267, 6.26571364526818396325045334870, 7.09678656437709992007811907801, 7.791900965553101706756931869572, 8.36369631467476814632289342130, 9.01465302568475545041127046056, 9.56878848903546909238377179680, 10.40050284866216416081778795540, 11.42380066937703459883302716149, 12.24153382317376877076304403256, 12.41969510566568601772939552123, 12.70510236664407295974998393568, 13.87256325377312956912695149307, 14.57615225682636254583733678286, 15.25863773287638270211060594587, 16.23091775109536160976518003139, 16.77760888596838496003297673951, 17.63551497769414855640118782932, 18.2983989534549899525510944433, 18.88486411845530803600695738902

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