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Properties

Label 1-4033-4033.1050-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.169 + 0.985i$
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 + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (0.802 − 0.597i)5-s + (−0.973 − 0.230i)6-s + (0.396 − 0.918i)7-s + (−0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.396 − 0.918i)13-s + (−0.835 − 0.549i)14-s + (−0.549 − 0.835i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (0.802 − 0.597i)5-s + (−0.973 − 0.230i)6-s + (0.396 − 0.918i)7-s + (−0.5 + 0.866i)8-s + (−0.993 − 0.116i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.396 − 0.918i)13-s + (−0.835 − 0.549i)14-s + (−0.549 − 0.835i)15-s + (0.766 + 0.642i)16-s + (−0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1050, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.169 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.268193747 - 1.068406177i\)
\(L(\frac12)\) \(\approx\) \(-1.268193747 - 1.068406177i\)
\(L(1)\) \(\approx\) \(0.3790251180 - 1.129128354i\)
\(L(1)\) \(\approx\) \(0.3790251180 - 1.129128354i\)

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 \( 1 + (0.0581 - 0.998i)T \)
5 \( 1 + (0.802 - 0.597i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (0.448 - 0.893i)T \)
13 \( 1 + (0.396 - 0.918i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.998 + 0.0581i)T \)
31 \( 1 + (-0.116 + 0.993i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.727 + 0.686i)T \)
53 \( 1 + (0.116 + 0.993i)T \)
59 \( 1 + (-0.835 + 0.549i)T \)
61 \( 1 + (-0.957 + 0.286i)T \)
67 \( 1 + (-0.802 - 0.597i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.893 - 0.448i)T \)
79 \( 1 + (0.396 + 0.918i)T \)
83 \( 1 + (0.0581 - 0.998i)T \)
89 \( 1 + (0.957 - 0.286i)T \)
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

−18.77122288703814806325230716235, −17.994740084398551862024881186062, −17.48276440821028152389940802135, −17.080099380271288077488975133688, −16.03287069557128948246229567815, −15.55582965486615285462805489751, −14.93646124340921333136046456299, −14.517407181041364910822987617541, −13.74008151222186474004140143416, −13.24911755894916922322452821620, −11.995755916988827379238554469821, −11.54467283627682181268762078454, −10.55123145145152253489180694276, −9.72374101018461449485143469325, −9.23036080846978111678662402688, −8.791190575258126852008164089683, −7.86369315440588731530451479662, −6.847059106343823682289891739747, −6.333181083666109286900047291144, −5.53346499241395316412588428068, −4.988152507867483013059217677228, −4.20773916503359905454101550081, −3.46931768793464600702422980180, −2.47001692681201987570319896789, −1.66065657843800032743749952440, 0.430308721429359409514305197356, 1.26829802244784494510471400087, 1.52397633401971400772471607226, 2.83756850318638672502789876063, 3.18294763464017591794035542971, 4.45041164350929620143719701969, 5.01751214000414406580390041391, 5.9995435047680609860132192339, 6.407427233825348649075699813139, 7.635277704754781338085537553574, 8.357324392783378515087786648011, 8.83344356086940378737645905151, 9.699660956397065293880242612347, 10.571793922147839860989315674361, 11.014954622639252922861420908169, 11.94020689523569535027047946652, 12.42048646447583691792415572296, 13.33785395480881614034031528771, 13.582416753354059117183659335571, 14.13044417654798091192175089679, 14.74138272743949577622741293178, 16.207300109233531921172924546123, 16.76384512011352262009432870077, 17.63896686721436423976464408489, 18.02138091438739679552816564769

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