Properties

Label 1-4033-4033.42-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.989 - 0.143i$
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  + (−0.173 + 0.984i)2-s + (−0.597 + 0.802i)3-s + (−0.939 − 0.342i)4-s + (−0.957 − 0.286i)5-s + (−0.686 − 0.727i)6-s + (−0.686 − 0.727i)7-s + (0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.835 − 0.549i)12-s + (−0.973 + 0.230i)13-s + (0.835 − 0.549i)14-s + (0.802 − 0.597i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.597 + 0.802i)3-s + (−0.939 − 0.342i)4-s + (−0.957 − 0.286i)5-s + (−0.686 − 0.727i)6-s + (−0.686 − 0.727i)7-s + (0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.835 − 0.549i)12-s + (−0.973 + 0.230i)13-s + (0.835 − 0.549i)14-s + (0.802 − 0.597i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.989 - 0.143i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.989 - 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5013460366 - 0.03606459078i\)
\(L(\frac12)\) \(\approx\) \(0.5013460366 - 0.03606459078i\)
\(L(1)\) \(\approx\) \(0.4822976105 + 0.1899471071i\)
\(L(1)\) \(\approx\) \(0.4822976105 + 0.1899471071i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (-0.957 - 0.286i)T \)
7 \( 1 + (-0.686 - 0.727i)T \)
11 \( 1 + (-0.448 - 0.893i)T \)
13 \( 1 + (-0.973 + 0.230i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.448 + 0.893i)T \)
31 \( 1 + (0.918 + 0.396i)T \)
41 \( 1 + (-0.642 + 0.766i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.230 - 0.973i)T \)
53 \( 1 + (0.549 + 0.835i)T \)
59 \( 1 + (-0.597 - 0.802i)T \)
61 \( 1 + (-0.448 + 0.893i)T \)
67 \( 1 + (0.998 + 0.0581i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.0581 - 0.998i)T \)
79 \( 1 + (-0.893 + 0.448i)T \)
83 \( 1 + (0.286 - 0.957i)T \)
89 \( 1 + (0.918 - 0.396i)T \)
97 \( 1 + (0.802 + 0.597i)T \)
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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

−18.692361353987499554718215637844, −18.08782124337583044919086086258, −17.19291834326443515887967783116, −16.79653407236314463682065788070, −15.76405138999128818941202440610, −15.02551455541702828627620091046, −14.37076289264288414646089623436, −13.28662263643848216290518904333, −12.592621261466582487973150578158, −12.39230677555432953161780244860, −11.75751155988656491973819385562, −11.09936012628101229537937692510, −10.1585484854448989776389889749, −9.87286175546218174267650859292, −8.65857800826529940856594058628, −7.99969889132340739825889693325, −7.468627759745480575021712610911, −6.61894773678987162960964925702, −5.68147017969761832498629615173, −4.92122134009815841043858183581, −4.16654578045881644572488059312, −3.1416944499855302853858449948, −2.52127407534174746851488189344, −1.80024431298646201219718880417, −0.62543996261649773792219639847, 0.35384369796591136077171871390, 0.93430576217302978171747520465, 3.16399565358774354589194529210, 3.42068029857175772175574382040, 4.58537329985774614610486464925, 4.917338285093308929745452492379, 5.604311891651568061621029208043, 6.71863651906381901971059977059, 7.12332475272946204877532044646, 7.82456437950878407018707929943, 8.89048500517128920364487002049, 9.25929178160768905280169797984, 10.10657856332125923767848574544, 10.72307359084012083405520804260, 11.5619059462660125367761091784, 12.20029092632948597773678157692, 13.199532444971373583609775986, 13.72942903975964980287079889340, 14.632535311157482809780216460796, 15.38054744734900330335093637368, 15.81333827324751238308324466518, 16.44031260279845711986286505399, 16.814254932998302163558844340343, 17.42291212923414027171838772498, 18.40476419879136557826931834652

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