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Properties

Label 1-4033-4033.2754-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.394 - 0.919i$
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  + (0.939 + 0.342i)2-s + (−0.0581 + 0.998i)3-s + (0.766 + 0.642i)4-s + (−0.993 + 0.116i)5-s + (−0.396 + 0.918i)6-s + (0.597 + 0.802i)7-s + (0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (−0.973 − 0.230i)10-s + (−0.973 + 0.230i)11-s + (−0.686 + 0.727i)12-s + (−0.396 + 0.918i)13-s + (0.286 + 0.957i)14-s + (−0.0581 − 0.998i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.0581 + 0.998i)3-s + (0.766 + 0.642i)4-s + (−0.993 + 0.116i)5-s + (−0.396 + 0.918i)6-s + (0.597 + 0.802i)7-s + (0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (−0.973 − 0.230i)10-s + (−0.973 + 0.230i)11-s + (−0.686 + 0.727i)12-s + (−0.396 + 0.918i)13-s + (0.286 + 0.957i)14-s + (−0.0581 − 0.998i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.394 - 0.919i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.394 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9958422691 + 1.510499654i\)
\(L(\frac12)\) \(\approx\) \(-0.9958422691 + 1.510499654i\)
\(L(1)\) \(\approx\) \(0.8282945393 + 1.127240838i\)
\(L(1)\) \(\approx\) \(0.8282945393 + 1.127240838i\)

Euler product

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

−18.30216210019643631420424303268, −17.14135936440566186209881643030, −16.741372332871734007697489203993, −15.79422830709353021321686315308, −14.99600549521730182187348207650, −14.62351860822389797592961434305, −13.66824561520415441682958600557, −13.24401365939200413469326690144, −12.39827808788296153044543652654, −12.16302354634966103293795481633, −11.14688446546537674869875125214, −10.78158985663703081715863646610, −10.111456610833455997824307258378, −8.61943954585164717691547298675, −7.976936109834630994488274293233, −7.37398261289386314254511723006, −6.93253729284017039841865009959, −5.837062169883632411656909799136, −5.14066676170317950079487295109, −4.56278002332504195657041406027, −3.51279798244966582458961027816, −2.93945688687508056901024236746, −2.098062550147427690180757112091, −1.01657187031300340859542360714, −0.38433215626077911726107416633, 1.70654224426756726272344283268, 2.76906297122630202360669315356, 3.206482772501588786924974724627, 4.21300196251320909924978301981, 4.71806309787904405988035543211, 5.2501120460031937748027998023, 6.0359537058313195624155583084, 7.01373168293500847863756527870, 7.73577027383967292797333229826, 8.500163752466821443497482095875, 8.948145339503176618917571837652, 10.30413491403303638641564072443, 10.74017963393132822858561555696, 11.625082114719684026068782401594, 11.96058801230282400039870229779, 12.65803698056413930593914551415, 13.60162797657444108086922402934, 14.64251888254777138102440312737, 14.880828298752523150993399544668, 15.28092647704953130054033748695, 16.20642777087831838792321968355, 16.45991623726478700209982132318, 17.30689250181225753150657548341, 18.18263228586416328653087708941, 19.13448855573747257920713487063

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