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Properties

Label 1-4033-4033.411-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.206 - 0.978i$
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  + (0.173 − 0.984i)2-s + (−0.686 − 0.727i)3-s + (−0.939 − 0.342i)4-s + (0.0581 + 0.998i)5-s + (−0.835 + 0.549i)6-s + (0.893 + 0.448i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.993 + 0.116i)10-s + (0.993 − 0.116i)11-s + (0.396 + 0.918i)12-s + (−0.835 + 0.549i)13-s + (0.597 − 0.802i)14-s + (0.686 − 0.727i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.686 − 0.727i)3-s + (−0.939 − 0.342i)4-s + (0.0581 + 0.998i)5-s + (−0.835 + 0.549i)6-s + (0.893 + 0.448i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.993 + 0.116i)10-s + (0.993 − 0.116i)11-s + (0.396 + 0.918i)12-s + (−0.835 + 0.549i)13-s + (0.597 − 0.802i)14-s + (0.686 − 0.727i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.206 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.256394534 - 1.018878112i\)
\(L(\frac12)\) \(\approx\) \(1.256394534 - 1.018878112i\)
\(L(1)\) \(\approx\) \(0.9000706798 - 0.4958228321i\)
\(L(1)\) \(\approx\) \(0.9000706798 - 0.4958228321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (-0.686 - 0.727i)T \)
5 \( 1 + (0.0581 + 0.998i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.993 - 0.116i)T \)
13 \( 1 + (-0.835 + 0.549i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (0.993 + 0.116i)T \)
31 \( 1 + (0.286 + 0.957i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.0581 - 0.998i)T \)
53 \( 1 + (-0.396 - 0.918i)T \)
59 \( 1 + (-0.686 + 0.727i)T \)
61 \( 1 + (-0.396 - 0.918i)T \)
67 \( 1 + (0.993 - 0.116i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.396 - 0.918i)T \)
79 \( 1 + (0.597 + 0.802i)T \)
83 \( 1 + (-0.835 + 0.549i)T \)
89 \( 1 + (0.686 + 0.727i)T \)
97 \( 1 + (-0.973 + 0.230i)T \)
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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

−18.08593712996281882195900895545, −17.52130876309622381904162015945, −17.14625338668375395868125285226, −16.76467994774582482839602507038, −15.89095434254397023446275499350, −15.36799632137898616667453746645, −14.61394432590360896268294889941, −14.117402496712627513800442992778, −13.20716481875228029292080813495, −12.40251067810628941960670795076, −11.90593142628423662446256379636, −11.14283381117019964294568372358, −10.05127629189271529449276678054, −9.54283899930107450782839359615, −8.95070335366432042178854228163, −7.95436830379412853817210978574, −7.60171565959219193450959609422, −6.46452950125631049764140587953, −5.789068583547586543289246212425, −5.17355622473234140863949254645, −4.475992017575458480528282391098, −4.140274390726486995639413464733, −3.15301076281563305887682598752, −1.397261232007934223683826535680, −0.805116545687943772997297907545, 0.71142892600351486204383723216, 1.58085720134223525950517334197, 2.35417085715722736576745009747, 2.89417930731908470082890962858, 4.00634589185433263416288719090, 4.95437552601455933220113344393, 5.33425860921760467686328916325, 6.42340192607156645020430259597, 6.96492844721699068923487679559, 7.807444290885602611470906979489, 8.67199265544853770097212235081, 9.50101906137391263877597498718, 10.257216171834575723199418001630, 11.016917213692122536346734825474, 11.51679468511607196483645250828, 12.10071737237713394434478421128, 12.37995497032525072871169125306, 13.75646868905948975662086367273, 14.096947418294754043487831717048, 14.46211745851343613670549913235, 15.450933439506255334957177612337, 16.52710458951437633910750770199, 17.32699780090007162765067354259, 17.89354564582629283137986229240, 18.352338012813722241845828009988

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