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Properties

Label 1-4033-4033.1557-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.499 - 0.866i$
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 + (−0.993 − 0.116i)3-s + (0.173 + 0.984i)4-s + (−0.973 + 0.230i)5-s + (−0.686 − 0.727i)6-s + (−0.286 + 0.957i)7-s + (−0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (−0.893 − 0.448i)10-s + (−0.893 + 0.448i)11-s + (−0.0581 − 0.998i)12-s + (−0.686 − 0.727i)13-s + (−0.835 + 0.549i)14-s + (0.993 − 0.116i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.993 − 0.116i)3-s + (0.173 + 0.984i)4-s + (−0.973 + 0.230i)5-s + (−0.686 − 0.727i)6-s + (−0.286 + 0.957i)7-s + (−0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (−0.893 − 0.448i)10-s + (−0.893 + 0.448i)11-s + (−0.0581 − 0.998i)12-s + (−0.686 − 0.727i)13-s + (−0.835 + 0.549i)14-s + (0.993 − 0.116i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.499 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.499 - 0.866i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.499 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2656264167 + 0.1533649967i\)
\(L(\frac12)\) \(\approx\) \(-0.2656264167 + 0.1533649967i\)
\(L(1)\) \(\approx\) \(0.5590783754 + 0.5186683818i\)
\(L(1)\) \(\approx\) \(0.5590783754 + 0.5186683818i\)

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

−18.15421976198466890754739778600, −16.8529600835825303610515082218, −16.54518825699179414197073427706, −15.92380521019331393736383473096, −15.2303915357444792101374917001, −14.417474534537753966793998605254, −13.6435791257308297607345843097, −12.90855689318922816353487056194, −12.40718275042407686763481391601, −11.626548028541818598235186640896, −11.21184795278354124187836821027, −10.57643115982550455808576164140, −9.84801489176930226393546453737, −9.227885895506750396771195596580, −7.8163631743237420836026074737, −7.18382175773280056022197611341, −6.6762104401371706401231070293, −5.48939404110463197498757703849, −5.0575834910027900212908661485, −4.39978482297376829266971568924, −3.599967377836633741370747327271, −3.05214910559864263665912393090, −1.72485977244452549754455133225, −0.72635484276397513519763565497, −0.107625659104316306738539693668, 1.52875450453484213914042002400, 2.85853995723423588738353238414, 3.24181998579226781440733993435, 4.327551919561407553055299889475, 5.21095882485057477449403586009, 5.38926004190255420920629701797, 6.27895531968847636545553752257, 7.15843662808061504324865637006, 7.645605424432101492691070376120, 8.17891634046133636945617122874, 9.344387008918951225776098296551, 10.21228329372199046408778807751, 11.048384312326138253339059539384, 11.8140750872807305558641686313, 12.15616839451397918371604591145, 12.88105189863677471470563735550, 13.24505134226067667555241605386, 14.637675280241438643190414377339, 15.100446990512608174895796344, 15.57726271143287326753646630339, 16.208258359053008153284755382798, 16.76210437863729494625113098252, 17.65837946611052747038871250703, 18.213162298778572427472371420799, 18.79917131745357717482687028935

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