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Properties

Label 1-4033-4033.3876-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: $\chi_{4033} (3876, \cdot )$
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

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

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