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Properties

Label 1-4033-4033.3125-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.979 - 0.201i$
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  − i·2-s + (0.993 + 0.116i)3-s − 4-s + (0.727 − 0.686i)5-s + (0.116 − 0.993i)6-s + (0.973 − 0.230i)7-s + i·8-s + (0.973 + 0.230i)9-s + (−0.686 − 0.727i)10-s + (0.686 − 0.727i)11-s + (−0.993 − 0.116i)12-s + (0.727 − 0.686i)13-s + (−0.230 − 0.973i)14-s + (0.802 − 0.597i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯
L(s)  = 1  − i·2-s + (0.993 + 0.116i)3-s − 4-s + (0.727 − 0.686i)5-s + (0.116 − 0.993i)6-s + (0.973 − 0.230i)7-s + i·8-s + (0.973 + 0.230i)9-s + (−0.686 − 0.727i)10-s + (0.686 − 0.727i)11-s + (−0.993 − 0.116i)12-s + (0.727 − 0.686i)13-s + (−0.230 − 0.973i)14-s + (0.802 − 0.597i)15-s + 16-s + (−0.866 − 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.979 - 0.201i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.979 - 0.201i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5497945852 - 5.407462455i\)
\(L(\frac12)\) \(\approx\) \(0.5497945852 - 5.407462455i\)
\(L(1)\) \(\approx\) \(1.383228538 - 1.422094529i\)
\(L(1)\) \(\approx\) \(1.383228538 - 1.422094529i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (0.993 + 0.116i)T \)
5 \( 1 + (0.727 - 0.686i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (0.686 - 0.727i)T \)
13 \( 1 + (0.727 - 0.686i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (-0.230 - 0.973i)T \)
31 \( 1 + (0.998 + 0.0581i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.993 + 0.116i)T \)
53 \( 1 + (0.396 + 0.918i)T \)
59 \( 1 + (-0.802 + 0.597i)T \)
61 \( 1 + (0.448 - 0.893i)T \)
67 \( 1 + (0.993 - 0.116i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.973 - 0.230i)T \)
79 \( 1 + (-0.802 + 0.597i)T \)
83 \( 1 + (-0.835 + 0.549i)T \)
89 \( 1 + (0.957 + 0.286i)T \)
97 \( 1 + (0.998 - 0.0581i)T \)
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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

−18.566719230561455462366412709169, −17.86037835849808970620553561576, −17.40346057378500209911559439715, −16.59429469203550419532339762262, −15.676766579612299862770409800259, −14.96411761844049329628489015288, −14.40816662257041578492634140143, −14.37489024667763281873393427466, −13.259103022485288482481355814171, −12.964455541984537196100454167541, −11.82892062928415032622066752009, −10.88033662168536643379561234360, −10.070249103378263797429758249168, −9.36690244995039300708447377402, −8.759180416244925809423612295366, −8.22258915524318892857135870998, −7.39376722697344882632463193287, −6.59339356446541295784770846200, −6.35082278068526257522463583523, −5.14007720499948076836946248678, −4.40302130714260829903400844050, −3.765174213412117301096442838445, −2.7638976210091591446247795068, −1.65934515674373368731496135019, −1.38942409711009646854940814361, 0.68424579021612798559875961980, 1.12269823867620801174927434263, 2.04588856045635065521447439995, 2.629227699720512373631031076035, 3.55190636316010931525495275729, 4.34918822181126870613908054236, 4.86637805094545985732775477827, 5.7012000040256388515692838226, 6.79317374272075490160520082465, 7.902775126165713295346830558472, 8.61426384676303233227150865089, 8.88036980995032673168483960391, 9.55397579232504030472714515796, 10.44058744300007063027924304, 11.03336108468299997040125240179, 11.69176846926998015466450105870, 12.68690759036646677680496543329, 13.35393258418401238923481682074, 13.73871112589883870438615537905, 14.18460011788328539559453735392, 15.17267292444118710908235334527, 15.73848805855852766718071326458, 17.02178628831556859531501406827, 17.34094649690996262671795183331, 18.094059806698712870757433656103

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