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Properties

Label 1-4033-4033.637-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.972 + 0.233i$
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 + (0.939 − 0.342i)3-s + (−0.5 + 0.866i)4-s + (−0.642 + 0.766i)5-s + (−0.766 − 0.642i)6-s + (0.766 + 0.642i)7-s + 8-s + (0.766 − 0.642i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s + (−0.173 + 0.984i)12-s + (0.766 + 0.642i)13-s + (0.173 − 0.984i)14-s + (−0.342 + 0.939i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.939 − 0.342i)3-s + (−0.5 + 0.866i)4-s + (−0.642 + 0.766i)5-s + (−0.766 − 0.642i)6-s + (0.766 + 0.642i)7-s + 8-s + (0.766 − 0.642i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s + (−0.173 + 0.984i)12-s + (0.766 + 0.642i)13-s + (0.173 − 0.984i)14-s + (−0.342 + 0.939i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.972 + 0.233i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.972 + 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.008740307 + 0.2378713148i\)
\(L(\frac12)\) \(\approx\) \(2.008740307 + 0.2378713148i\)
\(L(1)\) \(\approx\) \(1.181374890 - 0.1526528105i\)
\(L(1)\) \(\approx\) \(1.181374890 - 0.1526528105i\)

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 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.984 + 0.173i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.984 + 0.173i)T \)
31 \( 1 + (0.342 - 0.939i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.642 + 0.766i)T \)
53 \( 1 + (0.984 + 0.173i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (0.342 + 0.939i)T \)
97 \( 1 + (-0.342 - 0.939i)T \)
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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

−18.46109747662270836745694602420, −17.78188357036905290775433388065, −16.92045057456230505843145313335, −16.12700507446380612183923036919, −15.863092270986306781851403254215, −15.251356519257081190234991851694, −14.438643279299864324556569155198, −13.72140201514170239760547636166, −13.427595060532423947364177350649, −12.433359184197459998965445225832, −11.40101810589993318439000621191, −10.403248038453890910383600729789, −10.25769352418952580698689310332, −9.06347312865649202907775210705, −8.57650944480663691551910949933, −8.03028498837952603939075021425, −7.51716743257835154581066082380, −6.90053476288385868193855659577, −5.37478365267371452066702158163, −5.1222006768173051642769637733, −4.332458195297715769223325864644, −3.529422633196614213331081101540, −2.56345755865988081170543659969, −1.24391745828921144350435256016, −0.75077910886643206017124388286, 1.0797480235163335756284798899, 1.78196385501948161943341338561, 2.67410037765759688932808895076, 3.10943665332707957956095560332, 3.94460952914178173715228720533, 4.650654088324840367672502214445, 5.81013532358284715032549188437, 6.92090682671317203683315232874, 7.76351053583920788941765647357, 7.97752620677359177017147207397, 8.72305683549476016198125704605, 9.45914702802743289845724916442, 10.24184628096352529535240402977, 10.93875105931907505028302161553, 11.63839984588797886560004426164, 12.16819947082267479549145822455, 12.9525100386462403684518984043, 13.76073687783972877010928432628, 14.203145775855347393018487474677, 15.19287056786476551355357880015, 15.552488765899569284897686639416, 16.427008290709434346186052458190, 17.60325971201236689747919372802, 18.16527969362335077443059743391, 18.54797441164539908180549311288

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