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Properties

Label 1-4033-4033.2723-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.445 - 0.895i$
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  + (−0.939 + 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (0.866 − 0.5i)5-s + (−0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.642 + 0.766i)10-s + (0.642 + 0.766i)11-s + (−0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.766 + 0.642i)14-s − i·15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (0.866 − 0.5i)5-s + (−0.173 + 0.984i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.642 + 0.766i)10-s + (0.642 + 0.766i)11-s + (−0.173 − 0.984i)12-s + (−0.5 + 0.866i)13-s + (0.766 + 0.642i)14-s − i·15-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.445 - 0.895i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.445 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.362340775 - 0.8439036703i\)
\(L(\frac12)\) \(\approx\) \(1.362340775 - 0.8439036703i\)
\(L(1)\) \(\approx\) \(0.9374440629 - 0.2950213420i\)
\(L(1)\) \(\approx\) \(0.9374440629 - 0.2950213420i\)

Euler product

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

−18.74818087049172115602122902304, −17.71658034836820082354630676559, −17.39545087700330588427732107504, −16.70282204946749633786208072787, −15.73461771690473483832418681229, −15.39899099341932430533381703295, −14.77780422186949906011836950250, −13.74411433793903688707071190603, −13.20163357018686779675647044652, −12.31395976717617864688966185631, −11.33742914416765983709561837761, −10.90802466196741021425817185822, −10.11315863748351933467721068542, −9.5764019704474240113346607300, −8.988045128491854994320924166673, −8.55031211762776716407411503936, −7.59602565602355230609531973646, −6.6573302944179281033247804053, −5.97195373358448522928029306973, −5.28599437421819100524135302939, −4.05420986125650404984911267827, −3.18109934731719790618524708519, −2.63483938042961127370530317579, −2.16138938984615590711503349671, −0.822951932564947288600501501088, 0.77490000660434184944413402738, 1.333425706074496493533979890025, 2.2117352381224708922961268903, 2.76598211479969234732898326208, 4.12346573618057059542344172486, 4.94451139835217632762344585178, 6.12332390368259721577396979699, 6.646316052173043920159766230959, 7.05569666967291269579861935384, 7.83561644767548801814059417377, 8.775463373396246067293177636528, 9.208612911348685431301569304398, 9.83663453567512761712398053589, 10.44691362559799494278970337688, 11.504650536851355032931635644961, 12.30335148553495307015514445168, 12.77984502494947865060925845695, 13.86627451752253610861545911560, 14.18245513157019633184150270541, 14.744020069232319021396614361548, 15.94614782050598408531945394446, 16.46055922777195041428065130313, 17.25017940705568296059122307004, 17.5731071741790104729671525793, 18.22221515825563514892585232283

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