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Properties

Label 1-4033-4033.2234-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.745 + 0.666i$
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 + (0.984 − 0.173i)5-s + (−0.173 − 0.984i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (0.939 + 0.342i)12-s + (0.173 + 0.984i)13-s + (−0.939 − 0.342i)14-s + (−0.642 + 0.766i)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 + (0.984 − 0.173i)5-s + (−0.173 − 0.984i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (−0.342 + 0.939i)10-s + (0.342 + 0.939i)11-s + (0.939 + 0.342i)12-s + (0.173 + 0.984i)13-s + (−0.939 − 0.342i)14-s + (−0.642 + 0.766i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.745 + 0.666i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.745 + 0.666i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4971784320 + 1.301999121i\)
\(L(\frac12)\) \(\approx\) \(0.4971784320 + 1.301999121i\)
\(L(1)\) \(\approx\) \(0.6674825450 + 0.5694683823i\)
\(L(1)\) \(\approx\) \(0.6674825450 + 0.5694683823i\)

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 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (0.342 + 0.939i)T \)
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 \( 1 + (-0.342 + 0.939i)T \)
31 \( 1 + (0.642 - 0.766i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.984 - 0.173i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
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 \( 1 + (0.642 + 0.766i)T \)
97 \( 1 + (-0.642 - 0.766i)T \)
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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

−18.09567827588579133883325538110, −17.57802436456492231565870287580, −17.13287705435379951609695063298, −16.63834839627156277539615245893, −15.80083657564346120302224570355, −14.41591945425124937569797984001, −13.77852337827601044888165323760, −13.29537649652220723740378370054, −12.8044628189215063869233338472, −11.90453982149071668928620870609, −11.08681625138247534692316848519, −10.834857482650757054514688711403, −9.973752064220576838298465466106, −9.57372073729734589400660186438, −8.2524968674240499449236838451, −7.942968532092831688248337875691, −7.00851877351274918121690631723, −6.268625844145888741928450793313, −5.450195441460363497093752198962, −4.80767670903019018714124074552, −3.54087115248627624055134123046, −3.08927633651717631155165072643, −1.87302735968117937103623233305, −1.23331122459240452067812341606, −0.68252082471971374778664706633, 0.972993998759181736465065710561, 1.7409560248294885628089507338, 2.75347398864808402422006227812, 4.14483600927897969755333640032, 4.92056348891226414656718260773, 5.328264303629688645246410630239, 6.00663152368363507095846030694, 6.75081581042292192783325389323, 7.26333306912007929442306425462, 8.60067157780442356193904392953, 9.131185365098240621866933963, 9.634004025929511081855546620815, 10.108610288515979606471690054108, 11.05560714634164652919847779872, 11.83542572879765608953949761859, 12.45536944790359124721868857482, 13.431255429912800779815594628530, 14.33284574539677397123293004209, 14.68198296483830990769775031883, 15.508202959446523879320821413791, 16.15549090779446489057042310305, 16.697242084107351855052956572108, 17.28870320769672074355330024866, 17.96382385384114946095270568950, 18.42644583720743584693530717217

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