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Properties

Label 1-4033-4033.354-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0178 - 0.999i$
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  + (0.939 − 0.342i)2-s + 3-s + (0.766 − 0.642i)4-s + (0.5 − 0.866i)5-s + (0.939 − 0.342i)6-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + 9-s + (0.173 − 0.984i)10-s + (0.173 + 0.984i)11-s + (0.766 − 0.642i)12-s − 13-s + (−0.173 + 0.984i)14-s + (0.5 − 0.866i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + 3-s + (0.766 − 0.642i)4-s + (0.5 − 0.866i)5-s + (0.939 − 0.342i)6-s + (−0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + 9-s + (0.173 − 0.984i)10-s + (0.173 + 0.984i)11-s + (0.766 − 0.642i)12-s − 13-s + (−0.173 + 0.984i)14-s + (0.5 − 0.866i)15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0178 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0178 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0178 - 0.999i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (354, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0178 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.484700003 - 3.547299519i\)
\(L(\frac12)\) \(\approx\) \(3.484700003 - 3.547299519i\)
\(L(1)\) \(\approx\) \(2.415524294 - 1.061644989i\)
\(L(1)\) \(\approx\) \(2.415524294 - 1.061644989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−18.95738040411785466450554666713, −17.75765220987074157358286649700, −17.24631809435153259244976933893, −16.35611829776541415367053564175, −15.82072010382987856183304554149, −14.93860784751390608295507532162, −14.43367593538376398446248217364, −13.9922491250685576417081349855, −13.361672655845100363671412504695, −12.859161864881444287211007121009, −11.98359606625849946224783978513, −10.94932752496552863604229941470, −10.43351436036398486146017304667, −9.76526923614359455461553746485, −8.76044276807877749966021302362, −8.03829945658164619727383678206, −7.13352218105484171362238299817, −6.877524901441584161465008350152, −6.05954770432771806810918581639, −5.18390835513048829358622751147, −4.123700697118532272350859815637, −3.577515532106426261459105835885, −3.0022257209336287227218474246, −2.26336378664716027874839746888, −1.37964644397265026111927121565, 0.77815182812396299955790665289, 1.99740878093464101125959024450, 2.44129556578172172820590529202, 2.89057790822635142537206978351, 4.266852264089497641373169973875, 4.607320454536094165628941666026, 5.26133181061524316414108837404, 6.35551162666492420314647154811, 6.898860804350904262083198035487, 7.75650593064248638228720599037, 8.81136000956764892822786215097, 9.43453819118553758254038353028, 9.737493795983967349980797754675, 10.66993037183387968607295335313, 11.82595103173010500800737274437, 12.35858752832327077132231266396, 12.90739078489921459888363920997, 13.37538242320698793142597360024, 14.15647880665448297358419152000, 14.86573686001214463708387192540, 15.40845467858135612169993641510, 15.90362408301130443630832583048, 16.77687231925318349075102838261, 17.676861463116142563776395818854, 18.471162501920172402539507501598

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