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Properties

Label 1-4033-4033.198-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.993 + 0.109i$
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  + (0.866 + 0.5i)2-s + (−0.893 + 0.448i)3-s + (0.5 + 0.866i)4-s + (−0.116 − 0.993i)5-s + (−0.998 − 0.0581i)6-s + (0.597 + 0.802i)7-s + i·8-s + (0.597 − 0.802i)9-s + (0.396 − 0.918i)10-s + (−0.396 − 0.918i)11-s + (−0.835 − 0.549i)12-s + (−0.116 − 0.993i)13-s + (0.116 + 0.993i)14-s + (0.549 + 0.835i)15-s + (−0.5 + 0.866i)16-s + i·17-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.893 + 0.448i)3-s + (0.5 + 0.866i)4-s + (−0.116 − 0.993i)5-s + (−0.998 − 0.0581i)6-s + (0.597 + 0.802i)7-s + i·8-s + (0.597 − 0.802i)9-s + (0.396 − 0.918i)10-s + (−0.396 − 0.918i)11-s + (−0.835 − 0.549i)12-s + (−0.116 − 0.993i)13-s + (0.116 + 0.993i)14-s + (0.549 + 0.835i)15-s + (−0.5 + 0.866i)16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.993 + 0.109i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (198, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.993 + 0.109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1145562877 + 2.084839962i\)
\(L(\frac12)\) \(\approx\) \(0.1145562877 + 2.084839962i\)
\(L(1)\) \(\approx\) \(1.156614363 + 0.6316662190i\)
\(L(1)\) \(\approx\) \(1.156614363 + 0.6316662190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.893 + 0.448i)T \)
5 \( 1 + (-0.116 - 0.993i)T \)
7 \( 1 + (0.597 + 0.802i)T \)
11 \( 1 + (-0.396 - 0.918i)T \)
13 \( 1 + (-0.116 - 0.993i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.116 + 0.993i)T \)
31 \( 1 + (-0.727 + 0.686i)T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.0581 - 0.998i)T \)
53 \( 1 + (0.893 - 0.448i)T \)
59 \( 1 + (-0.549 - 0.835i)T \)
61 \( 1 + (-0.957 + 0.286i)T \)
67 \( 1 + (0.835 - 0.549i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.993 + 0.116i)T \)
79 \( 1 + (0.998 - 0.0581i)T \)
83 \( 1 + (0.973 + 0.230i)T \)
89 \( 1 + (0.116 + 0.993i)T \)
97 \( 1 + (-0.727 - 0.686i)T \)
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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

−18.101830198268584242333425027280, −17.408114154535962520250764416976, −16.61136736375465006583249965864, −15.78224467569621169141896667465, −15.181764059384562740320587809766, −14.32414702521040730976647800, −13.8209731889044547747043250900, −13.27565169616637080269576579141, −12.333804999548835747577497328, −11.738329428527738740560357520660, −11.17003312016822911226031419053, −10.73116913281725348267865502332, −10.01173640398991271500241724171, −9.286278786248776413627789697743, −7.546484962645925502918936338734, −7.31218426583622910325903536475, −6.74459430124592130699512187975, −5.94109954300125352243054405304, −4.98637761441787612973026697947, −4.5307123834475331153030815754, −3.82092484543445581207363169949, −2.527758201021648742140397504942, −2.18806418085635831050926312021, −1.09009307925548404907677562072, −0.27932119470597736362055664297, 0.983959594270709050766046262954, 1.83128174809490645451154496821, 3.224618642537423208027343590417, 3.665100151694385528339210465021, 4.77644249556311352301080011334, 5.277927926561969162318038380647, 5.624215318633211790447154387257, 6.26930670012044944996685363580, 7.44599591706363354351336744113, 8.14309068004144265272873909917, 8.69601287086685619322220957927, 9.55813816114290697467525368079, 10.749993032224994388416035553255, 11.10724062229221795542887112065, 12.05765804035205221452066446967, 12.41330591237829773790537245856, 12.998960698213472368120577361203, 13.80096529326331477293009865805, 14.832358648344806401426653910953, 15.26176607880787848439591949766, 15.92687462342046875759393499675, 16.50269148432883019243317541616, 16.97556010864664424717044804177, 17.84820579874672322982647456742, 18.196535733466273497878475427654

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