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Properties

Label 1-4033-4033.2-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.629 - 0.777i$
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 + (0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s − i·5-s + (−0.173 + 0.984i)6-s + 7-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.642 − 0.766i)10-s + (−0.642 − 0.766i)11-s + (−0.766 + 0.642i)12-s + (−0.5 − 0.866i)13-s + (0.766 + 0.642i)14-s + (0.866 − 0.5i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s − i·5-s + (−0.173 + 0.984i)6-s + 7-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.642 − 0.766i)10-s + (−0.642 − 0.766i)11-s + (−0.766 + 0.642i)12-s + (−0.5 − 0.866i)13-s + (0.766 + 0.642i)14-s + (0.866 − 0.5i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.629 - 0.777i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.629 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.268745439 - 0.6051274355i\)
\(L(\frac12)\) \(\approx\) \(1.268745439 - 0.6051274355i\)
\(L(1)\) \(\approx\) \(1.390941169 + 0.5637277998i\)
\(L(1)\) \(\approx\) \(1.390941169 + 0.5637277998i\)

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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 - iT \)
7 \( 1 + T \)
11 \( 1 + (-0.642 - 0.766i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.984 - 0.173i)T \)
31 \( 1 + (-0.642 - 0.766i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.642 - 0.766i)T \)
53 \( 1 + (0.984 - 0.173i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (-0.984 - 0.173i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (0.642 + 0.766i)T \)
97 \( 1 + (-0.984 - 0.173i)T \)
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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

−18.620752951651744697729315707768, −18.11149530711832972466705643119, −17.644512144904863140165921108188, −16.49742814196932879852298982721, −15.27959583572050025620215888802, −14.90047130886458952362487372506, −14.43274146465349680520708356753, −13.80625168712104277847749555911, −13.16151106655580795181879311907, −12.4466044215051613944491651944, −11.61166929281494934479638412496, −11.32543179279386528612392995407, −10.48676660802929108327961756791, −9.637579631427478673616330454057, −8.993629894982190640712246654219, −7.83210361136651042061973145021, −7.23416003795347449518074043948, −6.76562079591086990002986043049, −5.82827844433938209864824165387, −4.99060639630261741455339705135, −4.25271606548721089047892782289, −3.30574720894112185854615323598, −2.57783951890891116618016828904, −1.96740770143049000116049299912, −1.438075506661343155187936193154, 0.23103419569836986261314188844, 1.94649934176989559650428478455, 2.54295584267259795476818055438, 3.67649664128566822153514864293, 4.15794169377678031089437820212, 5.023150747915253817315138258301, 5.33211835185363190441252463041, 6.04896865129482645447479808428, 7.38416684873305911861210238433, 8.04572472608973172186670497651, 8.5617481737497561668046576531, 8.908728412339899320868899798443, 10.15814696701519606729524456761, 10.88411880314073881932543889577, 11.46366684170138096597383314346, 12.49584887375028062777823764818, 13.10343859923539807296799075246, 13.56045086531044639494222248396, 14.52776919699549563223060557339, 15.06192461093503929901525676534, 15.386349972434918576295848301013, 16.3184831458833373981706691367, 16.91457284321543813906425134886, 17.21370718990316198245398052470, 18.15728442531377868236143544563

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