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Properties

Label 1-4033-4033.1018-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.381 - 0.924i$
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 + (−0.396 − 0.918i)3-s + (0.173 + 0.984i)4-s + (0.727 − 0.686i)5-s + (−0.286 + 0.957i)6-s + (0.973 − 0.230i)7-s + (0.5 − 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.998 + 0.0581i)10-s + (0.998 + 0.0581i)11-s + (0.835 − 0.549i)12-s + (0.286 − 0.957i)13-s + (−0.893 − 0.448i)14-s + (−0.918 − 0.396i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.396 − 0.918i)3-s + (0.173 + 0.984i)4-s + (0.727 − 0.686i)5-s + (−0.286 + 0.957i)6-s + (0.973 − 0.230i)7-s + (0.5 − 0.866i)8-s + (−0.686 + 0.727i)9-s + (−0.998 + 0.0581i)10-s + (0.998 + 0.0581i)11-s + (0.835 − 0.549i)12-s + (0.286 − 0.957i)13-s + (−0.893 − 0.448i)14-s + (−0.918 − 0.396i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.381 - 0.924i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1018, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.381 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8940808064 - 1.336095334i\)
\(L(\frac12)\) \(\approx\) \(0.8940808064 - 1.336095334i\)
\(L(1)\) \(\approx\) \(0.7583748454 - 0.5805289400i\)
\(L(1)\) \(\approx\) \(0.7583748454 - 0.5805289400i\)

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 \( 1 + (-0.396 - 0.918i)T \)
5 \( 1 + (0.727 - 0.686i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (0.998 + 0.0581i)T \)
13 \( 1 + (0.286 - 0.957i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 - T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.998 - 0.0581i)T \)
31 \( 1 + (-0.802 + 0.597i)T \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.727 + 0.686i)T \)
53 \( 1 + (0.549 + 0.835i)T \)
59 \( 1 + (-0.396 + 0.918i)T \)
61 \( 1 + (-0.549 - 0.835i)T \)
67 \( 1 + (0.998 + 0.0581i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.835 + 0.549i)T \)
79 \( 1 + (-0.893 + 0.448i)T \)
83 \( 1 + (0.286 - 0.957i)T \)
89 \( 1 + (0.918 - 0.396i)T \)
97 \( 1 + (0.116 - 0.993i)T \)
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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

−18.51360507521546008043032912447, −17.83990018741668486827072375669, −17.24569770608933113734679982707, −16.71673451843217517407821138793, −16.226065545454779961327792643193, −15.13817750514613077484667930969, −14.716153554052037958715632324969, −14.322517966590076699755589202464, −13.67616466531786258184515973016, −12.21143731996482954262757411848, −11.40003421120293258686788584445, −11.01772893173124436544447348021, −10.341111161800547167925945013703, −9.57969811002028275778742299844, −9.02972733446894937120361515707, −8.514186152369161749336843776057, −7.42209511192631940999766119582, −6.5676626213999098943309923055, −6.19641044680828607754574785442, −5.31912498283161285490018927521, −4.69765307591472720371403364449, −3.86023521851256606478432274177, −2.63887591607770495035305303138, −1.82012090626802535667303895301, −0.89941403865638666138148417929, 0.864269435791936197281769262371, 1.27781362814276645123753467132, 1.89321071640172626955434730871, 2.78825018485374174153904396396, 3.87634858514495136878929736339, 4.770396029647863177138559719545, 5.63294789546742876185054491651, 6.34002162718547596807695306438, 7.248512013562689578449433801213, 7.90659024278454585555241816176, 8.61871642230150926502968097734, 8.99660326322460066918638225830, 10.16085645248846056226083589732, 10.68339326602660615265481193478, 11.34025164356664860936810895284, 12.18750962645837489048476563864, 12.57031566069805748857297938544, 13.23161265081222150918343724836, 13.974365385031026366476179060680, 14.6596294584087250973829965546, 15.776498262299206708243245725408, 16.83352293648481057912298083767, 17.16071287711867600909583669729, 17.48043752871467740298054230129, 18.1832631335156361301106996369

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