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Properties

Label 1-4033-4033.3975-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.997 - 0.0698i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.642 − 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 + 0.984i)4-s + (−0.835 − 0.549i)5-s + (−0.549 + 0.835i)6-s + (−0.0581 + 0.998i)7-s + (0.866 − 0.5i)8-s + (−0.835 + 0.549i)9-s + (0.116 + 0.993i)10-s + (0.116 − 0.993i)11-s + (0.993 − 0.116i)12-s + (0.448 + 0.893i)13-s + (0.802 − 0.597i)14-s + (−0.286 + 0.957i)15-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 + 0.984i)4-s + (−0.835 − 0.549i)5-s + (−0.549 + 0.835i)6-s + (−0.0581 + 0.998i)7-s + (0.866 − 0.5i)8-s + (−0.835 + 0.549i)9-s + (0.116 + 0.993i)10-s + (0.116 − 0.993i)11-s + (0.993 − 0.116i)12-s + (0.448 + 0.893i)13-s + (0.802 − 0.597i)14-s + (−0.286 + 0.957i)15-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.997 - 0.0698i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3975, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.997 - 0.0698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02146545561 - 0.6142116586i\)
\(L(\frac12)\) \(\approx\) \(0.02146545561 - 0.6142116586i\)
\(L(1)\) \(\approx\) \(0.4410037120 - 0.3279200355i\)
\(L(1)\) \(\approx\) \(0.4410037120 - 0.3279200355i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.642 - 0.766i)T \)
3 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (-0.835 - 0.549i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (0.116 - 0.993i)T \)
13 \( 1 + (0.448 + 0.893i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.993 - 0.116i)T \)
31 \( 1 + (0.286 - 0.957i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.998 + 0.0581i)T \)
53 \( 1 + (0.116 + 0.993i)T \)
59 \( 1 + (-0.957 - 0.286i)T \)
61 \( 1 + (-0.597 - 0.802i)T \)
67 \( 1 + (0.802 + 0.597i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.396 + 0.918i)T \)
79 \( 1 + (-0.918 + 0.396i)T \)
83 \( 1 + (0.0581 - 0.998i)T \)
89 \( 1 + (0.973 + 0.230i)T \)
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

−18.416923825473969431395045746468, −17.6655651547772821238886052512, −17.293674035502452695786543947349, −16.562370082727259001311504371410, −15.907081846441509034992791092448, −15.21270382095989950516184453987, −14.96008189631533789154637502626, −14.26355537731452006902308040207, −13.31814952536552344062870674902, −12.409448314373110511252982098228, −11.40263315196663460205199629287, −10.66036567112503340098530236966, −10.422515137728141168182526369803, −9.83453819472097678671350770531, −8.69505834515127673350770349813, −8.29758887574166764761596411763, −7.42894794690053759329057389875, −6.66950939971785490385169490389, −6.25795712787902395583786466220, −5.018982512526015973893163039485, −4.53645791880094053190695986726, −3.80483811036303674270743073629, −3.00922141577054803501128769824, −1.6127012663336786772039215645, −0.55756507804782770664104775161, 0.24277776607284426623876886452, 0.91747308920609831454483845774, 1.80932627645046420539750946823, 2.65009578593339188319197485628, 3.26361716406089695811867617720, 4.422106806901188722886370769922, 5.01592089235564689639235448900, 6.24271406544879943833863977737, 6.75080419171532866433066902956, 7.69803230623003846129175092121, 8.42370649260099237512255935911, 8.82786919438597129316542672106, 9.310340579229414283182615843657, 10.71363470422033452169323491981, 11.46031027933127468522884955642, 11.540775221241261609754954815299, 12.31559707463139534424188204594, 13.03661589900181889493093991461, 13.46283885977215394517995057662, 14.3909637712927675861292534380, 15.57617638316850122969441181424, 16.055145164901155121316077934572, 16.87280872484649084858281407378, 17.276758712337151294871336845372, 18.33983975254486996508405294855

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