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Properties

Label 1-4033-4033.3546-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0775 + 0.996i$
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 + (−0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (0.230 − 0.973i)5-s + (−0.835 − 0.549i)6-s + (−0.286 − 0.957i)7-s + (0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.918 + 0.396i)10-s + (−0.918 + 0.396i)11-s + (0.686 − 0.727i)12-s + (0.286 + 0.957i)13-s + (0.993 − 0.116i)14-s + (0.802 + 0.597i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.396 + 0.918i)3-s + (−0.939 − 0.342i)4-s + (0.230 − 0.973i)5-s + (−0.835 − 0.549i)6-s + (−0.286 − 0.957i)7-s + (0.5 − 0.866i)8-s + (−0.686 − 0.727i)9-s + (0.918 + 0.396i)10-s + (−0.918 + 0.396i)11-s + (0.686 − 0.727i)12-s + (0.286 + 0.957i)13-s + (0.993 − 0.116i)14-s + (0.802 + 0.597i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0775 + 0.996i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3546, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0775 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7108580354 + 0.6577223041i\)
\(L(\frac12)\) \(\approx\) \(0.7108580354 + 0.6577223041i\)
\(L(1)\) \(\approx\) \(0.6582732515 + 0.3595746206i\)
\(L(1)\) \(\approx\) \(0.6582732515 + 0.3595746206i\)

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 \( 1 + (-0.396 + 0.918i)T \)
5 \( 1 + (0.230 - 0.973i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (-0.918 + 0.396i)T \)
13 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.802 - 0.597i)T \)
31 \( 1 + (0.957 + 0.286i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.448 + 0.893i)T \)
53 \( 1 + (-0.230 + 0.973i)T \)
59 \( 1 + (-0.597 + 0.802i)T \)
61 \( 1 + (-0.448 - 0.893i)T \)
67 \( 1 + (-0.957 - 0.286i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.396 - 0.918i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (0.993 - 0.116i)T \)
89 \( 1 + (0.549 - 0.835i)T \)
97 \( 1 + (0.727 + 0.686i)T \)
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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

−18.24117956160500839754220409720, −18.07414382001439117757574374987, −17.43624812162046818506613052003, −16.28395691407482203556820337780, −15.70456169968907830534152872844, −14.64893284574271162524642590895, −13.89329994067720461340791105533, −13.46021050246866233614026847901, −12.72460838856943279583600980442, −11.84951906274295243237904734533, −11.76310060238358838153043561504, −10.67674356398348522883047489240, −10.24727890254499526519048425241, −9.53407502943341011885954987726, −8.413708996559784354740310668153, −7.95859769202504923114100882673, −7.25305573710475211079249205428, −6.16272639574257554279419093408, −5.54219051037509578511227473479, −5.10193117915014286652456781020, −3.48195355139820729512354559103, −2.94689548231026450745661312821, −2.468214467976232537909820829573, −1.567258146373131915460191721910, −0.51908959148761055251686085239, 0.63683688767310343772798562786, 1.49230936941197341943016383873, 3.08489281393908023788319618229, 4.06381146624465855632756411258, 4.58624364785378927126193162482, 5.04507489565198130401654544133, 6.07285308268611718059614196352, 6.38674084481122638731167815665, 7.647654906691563458240927288530, 8.04991658283743253237538042316, 9.02851304301260731590433091195, 9.57986982577202220620388096001, 10.1780535643370855224914138231, 10.67023670545011798224137096606, 11.919050412326833332165236941660, 12.47135728197467082870239596926, 13.55863742749482077966146914898, 13.80806887476293344893094339792, 14.64038440242303854435863199544, 15.62333876240794604921075354160, 16.00404137540935271076296840952, 16.49767138527674115737021471399, 17.098569761718224721135006016399, 17.622832581578771343614713634944, 18.29752250397426859589040556743

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