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Properties

Label 1-4033-4033.994-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.765 + 0.643i$
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  − 2-s + (0.686 − 0.727i)3-s + 4-s + (−0.549 + 0.835i)5-s + (−0.686 + 0.727i)6-s + (−0.0581 + 0.998i)7-s − 8-s + (−0.0581 − 0.998i)9-s + (0.549 − 0.835i)10-s + (−0.549 − 0.835i)11-s + (0.686 − 0.727i)12-s + (0.835 + 0.549i)13-s + (0.0581 − 0.998i)14-s + (0.230 + 0.973i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.686 − 0.727i)3-s + 4-s + (−0.549 + 0.835i)5-s + (−0.686 + 0.727i)6-s + (−0.0581 + 0.998i)7-s − 8-s + (−0.0581 − 0.998i)9-s + (0.549 − 0.835i)10-s + (−0.549 − 0.835i)11-s + (0.686 − 0.727i)12-s + (0.835 + 0.549i)13-s + (0.0581 − 0.998i)14-s + (0.230 + 0.973i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.765 + 0.643i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (994, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.765 + 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1014193319 + 0.2783902023i\)
\(L(\frac12)\) \(\approx\) \(0.1014193319 + 0.2783902023i\)
\(L(1)\) \(\approx\) \(0.6581040212 + 0.008437261544i\)
\(L(1)\) \(\approx\) \(0.6581040212 + 0.008437261544i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.686 - 0.727i)T \)
5 \( 1 + (-0.549 + 0.835i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.549 - 0.835i)T \)
13 \( 1 + (0.835 + 0.549i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.998 + 0.0581i)T \)
31 \( 1 + (0.918 - 0.396i)T \)
41 \( 1 + (0.342 + 0.939i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.727 - 0.686i)T \)
53 \( 1 + (0.957 - 0.286i)T \)
59 \( 1 + (-0.973 + 0.230i)T \)
61 \( 1 + (0.116 + 0.993i)T \)
67 \( 1 + (0.727 - 0.686i)T \)
71 \( 1 - T \)
73 \( 1 + (0.0581 + 0.998i)T \)
79 \( 1 + (-0.973 + 0.230i)T \)
83 \( 1 + (-0.597 - 0.802i)T \)
89 \( 1 + (0.448 + 0.893i)T \)
97 \( 1 + (-0.918 - 0.396i)T \)
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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

−18.08750558455971387012043537018, −17.31329952524697205211955731306, −17.01225271112288582098639252994, −15.98940141696511801165424714321, −15.63019848686255686578835783885, −15.28258816349025078683726477792, −14.214927707948074268033920818795, −13.36732384905557865175563810287, −12.76127770398440621443495547902, −11.91338190951349166127617715086, −10.90403355336891642020464009043, −10.48223799308609384910335713805, −9.96966628601594876921356919489, −9.05070822155651848170532327434, −8.46086515057705102138568878504, −7.99296021336836327316192892297, −7.347381494076356270472899688630, −6.44382920348260692100695062393, −5.39607511958577850534476356084, −4.40288915954499294016961186793, −3.97127742679769342781780730252, −3.044481141944500168111959943916, −2.08672120471872153289965786928, −1.26124035687788925390261260751, −0.112724666919900944121248814421, 1.08222147042823100348208708764, 2.18147514229162515794030078723, 2.65106716028933175440987687828, 3.25935637102313566682709448291, 4.24854325329072045466128937738, 5.859850968168734336871671812557, 6.297979099526692077379907833050, 6.880902517509513304445402177303, 7.80227452889145855469655806367, 8.455837274599405959902833103361, 8.61256703388266812244206938417, 9.65558368055595200803427403102, 10.36533187698672296244610296555, 11.28104924344867588229901735395, 11.74011493004332850462595358532, 12.315313827676429468812558705769, 13.32862479346428136256149147732, 14.0676208246253633662182793960, 14.768429911585578796700948432814, 15.47593680157837664025291852146, 15.931065129104070461356250624599, 16.61744961147563703837428663111, 17.964706922044315288385261992, 18.125301838593646942487218181439, 18.79613257480099790231575118027

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