Properties

Label 1-4033-4033.2808-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.820 + 0.571i$
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  − 2-s + (0.597 + 0.802i)3-s + 4-s + (0.973 + 0.230i)5-s + (−0.597 − 0.802i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (−0.973 − 0.230i)10-s + (−0.973 + 0.230i)11-s + (0.597 + 0.802i)12-s + (−0.973 − 0.230i)13-s + (0.286 + 0.957i)14-s + (0.396 + 0.918i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.597 + 0.802i)3-s + 4-s + (0.973 + 0.230i)5-s + (−0.597 − 0.802i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (−0.973 − 0.230i)10-s + (−0.973 + 0.230i)11-s + (0.597 + 0.802i)12-s + (−0.973 − 0.230i)13-s + (0.286 + 0.957i)14-s + (0.396 + 0.918i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.820 + 0.571i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2808, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.820 + 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2743717508 + 0.8744363408i\)
\(L(\frac12)\) \(\approx\) \(0.2743717508 + 0.8744363408i\)
\(L(1)\) \(\approx\) \(0.7719901127 + 0.2901702958i\)
\(L(1)\) \(\approx\) \(0.7719901127 + 0.2901702958i\)

Euler product

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

−18.36022494095854912997587617677, −17.807877887151349143857367565697, −16.97847671121141939608167903476, −16.319518663661097453026622578476, −15.46099014519378920268923334120, −14.9676041567553893806280445013, −13.97490833091104254739492976826, −13.47736634355444892442689890879, −12.597901418664119764658486666610, −12.01520651672032276351593372007, −11.479356174060018961157739150610, −10.178334674809773495244887824880, −9.78128232903391332821819747860, −9.06249178656050636582239904897, −8.667085661544700585658375099421, −7.6441769068307768413338647034, −7.24880900331395458933256548487, −6.391028072603750607510144372533, −5.551674964405468186271977273903, −5.15920297361048269473923930605, −3.20847469284811169700301648454, −2.801634131695174887684489795316, −2.119886030524022637386076933391, −1.4372904124340547734481261264, −0.31350882887644326922026767647, 1.13801648521446267298630810170, 2.11458273048230838427513473823, 2.80307332674932139454555024607, 3.38200072234050318619621717093, 4.56382127412249498838558614015, 5.3326994665057966764809163281, 6.19381728137703007700663068837, 6.99962527501380874752893749461, 7.93931601873854468871789323424, 8.11379790577249458048155809984, 9.36096261792500502014621929644, 9.84483520872402530453367786516, 10.28022429680717209691955070302, 10.603794612301837982598198627921, 11.62131764816731847359880117034, 12.69587217031073093383754450571, 13.31121064389378208818136487132, 14.233987115886694744097564058233, 14.67357805242079765998773197116, 15.459667898171445936341312266894, 16.169337010347394157559381338812, 16.894538237729685020073829078730, 17.17015411074181749446274152535, 18.06170441534803786422326100991, 18.70072050163975291859608606895

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