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Properties

Label 1-4033-4033.153-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.292 - 0.956i$
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 + (−0.973 + 0.230i)3-s + (0.173 + 0.984i)4-s + (−0.448 + 0.893i)5-s + (0.893 + 0.448i)6-s + (−0.835 + 0.549i)7-s + (0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.396 − 0.918i)12-s + (0.0581 + 0.998i)13-s + (0.993 + 0.116i)14-s + (0.230 − 0.973i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.973 + 0.230i)3-s + (0.173 + 0.984i)4-s + (−0.448 + 0.893i)5-s + (0.893 + 0.448i)6-s + (−0.835 + 0.549i)7-s + (0.5 − 0.866i)8-s + (0.893 − 0.448i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.396 − 0.918i)12-s + (0.0581 + 0.998i)13-s + (0.993 + 0.116i)14-s + (0.230 − 0.973i)15-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.292 - 0.956i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.292 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02685128583 + 0.03627570307i\)
\(L(\frac12)\) \(\approx\) \(0.02685128583 + 0.03627570307i\)
\(L(1)\) \(\approx\) \(0.3802602400 + 0.08346595311i\)
\(L(1)\) \(\approx\) \(0.3802602400 + 0.08346595311i\)

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 \( 1 + (-0.973 + 0.230i)T \)
5 \( 1 + (-0.448 + 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.918 - 0.396i)T \)
13 \( 1 + (0.0581 + 0.998i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.918 + 0.396i)T \)
31 \( 1 + (-0.230 + 0.973i)T \)
41 \( 1 + (-0.642 + 0.766i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.549 + 0.835i)T \)
53 \( 1 + (0.918 - 0.396i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (0.116 - 0.993i)T \)
67 \( 1 + (0.116 + 0.993i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.597 - 0.802i)T \)
79 \( 1 + (-0.597 - 0.802i)T \)
83 \( 1 + (0.835 - 0.549i)T \)
89 \( 1 + (-0.727 + 0.686i)T \)
97 \( 1 + (0.727 - 0.686i)T \)
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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

−17.911689264061882371891687542166, −17.10069285699297502293824144170, −16.81008438365102659937086610044, −15.98552848290485012571716148522, −15.64008795685257929631405171156, −15.073536137881521600041531519074, −13.49756081266058189551007360651, −13.37189401672944432948885673899, −12.50718454383360056255558770242, −11.714374966589682487588524372575, −11.01255673288763577987115898462, −10.24724573637182504647483486533, −9.70297861017556941926467558114, −9.06199212244341070149784176992, −7.84627911801650214380977992007, −7.52167244997338113806996545489, −7.02679252987355025353189834670, −5.80566657032350608301093049049, −5.48550243716745488033410850059, −4.82614870068899522794550629823, −3.84810235952176643612752716362, −2.65883179392340062739497118338, −1.422486562013933371876894763888, −0.62667870415854594141576123521, −0.033665585299191749369295051217, 1.28596219946299645368734732391, 2.2682642840390162547897062048, 3.240966114499444237747639886574, 3.63481287598993499768154046045, 4.62898844295598880841670552973, 5.74024989481129207367137656625, 6.36792879414240159343446391300, 7.07770188747749893406725730694, 7.76001004205766829490670974714, 8.65087660655169936046002399388, 9.489887925565265879896872613, 10.224598059755078162749995371444, 10.5675096258930767651479892516, 11.327514387466935506152977469547, 11.92769387429548404756100753677, 12.52508227291812912782302053973, 13.08750444237024061258882854612, 14.23269745986054176036652650503, 15.06246348598937339161502709728, 16.011044400933516696418582082, 16.24170974668746172715667291477, 16.81752540665441109544522971314, 17.84006296801281304025724881956, 18.42041918395448114836000628995, 18.94619267837330235889534552753

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