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Properties

Label 1-4033-4033.661-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.439 + 0.898i$
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 + (0.686 − 0.727i)3-s − 4-s + (−0.549 + 0.835i)5-s + (0.727 + 0.686i)6-s + (−0.0581 + 0.998i)7-s − i·8-s + (−0.0581 − 0.998i)9-s + (−0.835 − 0.549i)10-s + (0.835 − 0.549i)11-s + (−0.686 + 0.727i)12-s + (−0.549 + 0.835i)13-s + (−0.998 − 0.0581i)14-s + (0.230 + 0.973i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + i·2-s + (0.686 − 0.727i)3-s − 4-s + (−0.549 + 0.835i)5-s + (0.727 + 0.686i)6-s + (−0.0581 + 0.998i)7-s − i·8-s + (−0.0581 − 0.998i)9-s + (−0.835 − 0.549i)10-s + (0.835 − 0.549i)11-s + (−0.686 + 0.727i)12-s + (−0.549 + 0.835i)13-s + (−0.998 − 0.0581i)14-s + (0.230 + 0.973i)15-s + 16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.439 + 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.057468420 + 0.6596044271i\)
\(L(\frac12)\) \(\approx\) \(1.057468420 + 0.6596044271i\)
\(L(1)\) \(\approx\) \(0.8179233968 + 0.3964704963i\)
\(L(1)\) \(\approx\) \(0.8179233968 + 0.3964704963i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (0.686 - 0.727i)T \)
5 \( 1 + (-0.549 + 0.835i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (0.835 - 0.549i)T \)
13 \( 1 + (-0.549 + 0.835i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (-0.998 - 0.0581i)T \)
31 \( 1 + (-0.918 + 0.396i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.686 - 0.727i)T \)
53 \( 1 + (-0.286 - 0.957i)T \)
59 \( 1 + (-0.230 - 0.973i)T \)
61 \( 1 + (-0.116 - 0.993i)T \)
67 \( 1 + (0.686 + 0.727i)T \)
71 \( 1 + T \)
73 \( 1 + (0.0581 + 0.998i)T \)
79 \( 1 + (-0.230 - 0.973i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (0.448 + 0.893i)T \)
97 \( 1 + (-0.918 - 0.396i)T \)
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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

−18.29729061159425794115876974040, −17.420514430861721309958466554245, −16.78586321870385585883839454637, −16.35190310048894821585632939149, −15.2768900577428577402702792909, −14.63489358666655725524579553510, −14.13669284723172984042132860416, −13.15079038711178525625410687423, −12.80350977556739562450567802231, −12.05079119480502841748434561088, −11.03196000120800101826484303874, −10.73046103725794672975940705204, −9.71549396409831738394289883793, −9.4091711941962938533260990319, −8.63776218314819644932700052466, −7.867726371152189051713847264760, −7.38035708937438830510919890478, −5.94799102314912516180588964742, −4.88155898854428837340514433986, −4.324545110291148219404461081521, −4.01487888034977016687646989816, −3.16936101115617739599643447195, −2.22233385537600813121642905539, −1.41116482497976391528534428239, −0.40261947831441549009856533163, 0.35703949064487877390585767782, 1.79565131537538396821996527874, 2.42819540818898598213360691790, 3.64022384541462800948750969539, 3.8716288393142551821662780560, 5.10803585509580666274948469668, 6.09255888048489824186669736443, 6.61051557003878912892524335786, 7.09329027417606749345552188624, 7.88904907341040877472819294256, 8.603397316798880561083265860, 9.14846093712258885693711420735, 9.64928545119205201903218157615, 11.05087896473449648081536847348, 11.61518674131549651474989017484, 12.46756514467255215715239680220, 13.04917287020963943387197695645, 13.960592389841604373665480633565, 14.42031831932015567041785176617, 15.000940596852212018033447646438, 15.487170872192938712449512826214, 16.19436989178780864642090346671, 17.23798906763136109129987886551, 17.725869730924612759775062744744, 18.46474479696234787734545162093

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