Properties

Label 1-4033-4033.1564-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.777 - 0.628i$
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  + (−0.5 + 0.866i)2-s + (0.766 − 0.642i)3-s + (−0.5 − 0.866i)4-s + (0.173 + 0.984i)5-s + (0.173 + 0.984i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (−0.939 − 0.342i)12-s + (0.173 + 0.984i)13-s + (−0.939 − 0.342i)14-s + (0.766 + 0.642i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.766 − 0.642i)3-s + (−0.5 − 0.866i)4-s + (0.173 + 0.984i)5-s + (0.173 + 0.984i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (−0.939 − 0.342i)12-s + (0.173 + 0.984i)13-s + (−0.939 − 0.342i)14-s + (0.766 + 0.642i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.777 - 0.628i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1564, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.777 - 0.628i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2549831022 + 0.7213535416i\)
\(L(\frac12)\) \(\approx\) \(-0.2549831022 + 0.7213535416i\)
\(L(1)\) \(\approx\) \(0.7276347747 + 0.4985702413i\)
\(L(1)\) \(\approx\) \(0.7276347747 + 0.4985702413i\)

Euler product

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

−18.092335552146411534713962032063, −17.448967369455389810033299214482, −16.67125452051825277856464582115, −16.08424334750511643029752497127, −15.66667458380897932778737885271, −14.42092637045877935273769845955, −13.69002068033700236243251753450, −13.27820463517728495996769033072, −12.77780192358717170014869118137, −11.709669561928000876897456801007, −10.98095642851681326771263319800, −10.29281717863766745252210778705, −9.858771824353569945318890048739, −9.09943903075299828463422044978, −8.44431402851907294324463880365, −7.77256361364242307590535694825, −7.42464655785130665157847824847, −5.78565157908192419482509938895, −4.767455161405299885753113790059, −4.57767188333129436324750543331, −3.501747875134043961341375250166, −2.95281653257225987932750829996, −2.06879031630747028826465573904, −1.11087778541623565980729558923, −0.22563147134938357439187572621, 1.64044325238963782204707233013, 1.94887338578682488683804645601, 2.91767071895729224912643320642, 3.804933158460959851225616654, 4.90852790715483375715599705417, 5.84560113204783604162739675612, 6.31699060815542510155748387819, 7.15637796633846088353283204004, 7.62328779305678562841498618800, 8.384042821818927254472943277302, 8.99508324655504221833015608726, 9.746069831013620741232240438009, 10.28501846479900160574607958384, 11.383853888139902416577916756279, 11.93853351525759147019204465749, 13.12629643764798708256067150272, 13.559444984608445418762885693953, 14.27994669717636327652859020441, 14.87326012648867009412158349543, 15.47276016841535648801254473714, 15.81176376759570263906045429126, 17.00531225984506626184822623662, 17.87161970190154318926639536260, 18.16880883069046480136105676935, 18.71118981332835580857450057382

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