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Properties

Label 1-4033-4033.736-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.633 - 0.774i$
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  + (−0.766 + 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.939 − 0.342i)5-s + (0.939 − 0.342i)6-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.939 − 0.342i)10-s + (0.939 + 0.342i)11-s + (−0.5 + 0.866i)12-s + (−0.766 + 0.642i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.939 − 0.342i)5-s + (0.939 − 0.342i)6-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.939 − 0.342i)10-s + (0.939 + 0.342i)11-s + (−0.5 + 0.866i)12-s + (−0.766 + 0.642i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.633 - 0.774i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (736, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.633 - 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1701157674 - 0.08063228083i\)
\(L(\frac12)\) \(\approx\) \(0.1701157674 - 0.08063228083i\)
\(L(1)\) \(\approx\) \(0.3556896076 + 0.04203312356i\)
\(L(1)\) \(\approx\) \(0.3556896076 + 0.04203312356i\)

Euler product

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

−18.54274264970788068376296926099, −17.990774290355328657695645204103, −17.08663433654201443343937707477, −16.75074780269033645782567241059, −16.004153905908738093766883342437, −15.36391463138196869954240070887, −14.86562088365320405579267252857, −13.47354898515899636239030134031, −12.68220657303165291110708033428, −12.14310755462502333479374199332, −11.712625820539169157798510132, −10.93600557917956916902640375913, −10.37832148832433126669918567687, −9.72056842790253221591262992158, −9.00455907964423233094657435440, −8.244886993773137636579356528604, −7.35206509721619122889839808851, −6.62923564021784398868430855234, −6.17241396027080158950790483582, −4.92449765721955099963018908721, −4.068790283887584367545269726510, −3.50628307632296280653525019061, −2.7966737231349077287108412540, −1.62134898519013114204258993490, −0.456900846128845402373859864091, 0.18844258326331009613348449865, 1.21114735214835197844914466370, 2.02107585528630164380144594458, 3.41362593203488818974362647555, 4.49949354580407441276388921721, 4.80561211804457894559529762180, 6.05552821695567839195070290349, 6.48509465952665977416197698717, 7.19669839176394702659295091738, 7.66005019670469328740708107885, 8.524914251345979979830357707883, 9.586504315873635508388670543045, 9.80886005081104319979889867140, 10.75658396698714277233806099471, 11.59885931687749989193781008829, 12.04898806866679431211976248590, 12.63971124828947799876674236129, 13.77592024125175417961052813756, 14.30853463237400705557463164944, 15.332448274770781015943763165999, 15.96454707181910908041577549710, 16.42531411523594535725453482086, 16.92672577435109474309595107018, 17.49024719755988638837573469494, 18.3993057671996756321198765028

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