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Properties

Label 1-4033-4033.383-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.550 - 0.834i$
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 + (−0.973 − 0.230i)3-s + (−0.5 + 0.866i)4-s + (−0.998 − 0.0581i)5-s + (0.286 + 0.957i)6-s + (0.893 − 0.448i)7-s + 8-s + (0.893 + 0.448i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (0.686 − 0.727i)12-s + (−0.0581 + 0.998i)13-s + (−0.835 − 0.549i)14-s + (0.957 + 0.286i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.973 − 0.230i)3-s + (−0.5 + 0.866i)4-s + (−0.998 − 0.0581i)5-s + (0.286 + 0.957i)6-s + (0.893 − 0.448i)7-s + 8-s + (0.893 + 0.448i)9-s + (0.448 + 0.893i)10-s + (−0.448 + 0.893i)11-s + (0.686 − 0.727i)12-s + (−0.0581 + 0.998i)13-s + (−0.835 − 0.549i)14-s + (0.957 + 0.286i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.550 - 0.834i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.550 - 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7068960881 - 0.3803823829i\)
\(L(\frac12)\) \(\approx\) \(0.7068960881 - 0.3803823829i\)
\(L(1)\) \(\approx\) \(0.5596976037 - 0.2129252512i\)
\(L(1)\) \(\approx\) \(0.5596976037 - 0.2129252512i\)

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 \( 1 + (-0.973 - 0.230i)T \)
5 \( 1 + (-0.998 - 0.0581i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.448 + 0.893i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.549 - 0.835i)T \)
31 \( 1 + (0.802 + 0.597i)T \)
41 \( 1 + (-0.984 - 0.173i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.957 - 0.286i)T \)
53 \( 1 + (0.957 - 0.286i)T \)
59 \( 1 + (-0.286 + 0.957i)T \)
61 \( 1 + (-0.802 + 0.597i)T \)
67 \( 1 + (0.727 - 0.686i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.835 - 0.549i)T \)
79 \( 1 + (0.973 - 0.230i)T \)
83 \( 1 + (-0.597 - 0.802i)T \)
89 \( 1 + (-0.448 - 0.893i)T \)
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

−18.37940078741499636452007904018, −17.86212876391033554214784528149, −17.11440394980030098377884262934, −16.62100658240209760643908856203, −15.78090192752437730287838675680, −15.34960964119179457720803296704, −14.94009007364766609841367764448, −14.03537031224708350769007462528, −12.98009446059577732404549048669, −12.43507080100140780280292965247, −11.34680202758523089271484683664, −11.0339148020825623357655851489, −10.4548630622310180164172370608, −9.54742575786992370764691903388, −8.41822807700343152079968646081, −8.27036647293332323471297043470, −7.40892567870651401712286096981, −6.64927754094467856657782453464, −5.881404977818192649535052949232, −5.07181114067254185302977660408, −4.83599949392489042928275095810, −3.77653184930761202380776545832, −2.762847069381591754307546242075, −1.20628805413477889116229798030, −0.67040626765436250951032861384, 0.596350577882680085598982917585, 1.39391917966655353068528522519, 2.126840310283451459696445903748, 3.29628382871681776029068488212, 4.218118423352724170215577225470, 4.701863548009360866163672439255, 5.25717604259646088303030447256, 6.753935824996949801249570209160, 7.33725144907026702801702482876, 7.79881176531986996480604392938, 8.568584598817126185163339800212, 9.59148442148218899736913636781, 10.295463921882162510863304164626, 10.8498874474636410271189372880, 11.680551082114247284959155808248, 11.92174572911153044803317463894, 12.4313771380930752799459262912, 13.45448251330070550385895914527, 14.03580552574841246316619454408, 15.09577352743645294955664785750, 15.8450976015601643619104924854, 16.8020288839374722025819072657, 16.85344433896492046120604828579, 17.93512246764708639064801390274, 18.28253628974419771529284724436

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