Properties

Label 1-4033-4033.3653-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.816 - 0.577i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.984 − 0.173i)2-s + (0.893 − 0.448i)3-s + (0.939 + 0.342i)4-s + (−0.396 + 0.918i)5-s + (−0.957 + 0.286i)6-s + (−0.993 + 0.116i)7-s + (−0.866 − 0.5i)8-s + (0.597 − 0.802i)9-s + (0.549 − 0.835i)10-s + (0.549 + 0.835i)11-s + (0.993 − 0.116i)12-s + (−0.116 − 0.993i)13-s + (0.998 + 0.0581i)14-s + (0.0581 + 0.998i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (0.893 − 0.448i)3-s + (0.939 + 0.342i)4-s + (−0.396 + 0.918i)5-s + (−0.957 + 0.286i)6-s + (−0.993 + 0.116i)7-s + (−0.866 − 0.5i)8-s + (0.597 − 0.802i)9-s + (0.549 − 0.835i)10-s + (0.549 + 0.835i)11-s + (0.993 − 0.116i)12-s + (−0.116 − 0.993i)13-s + (0.998 + 0.0581i)14-s + (0.0581 + 0.998i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.816 - 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.676403036 - 0.5327563178i\)
\(L(\frac12)\) \(\approx\) \(1.676403036 - 0.5327563178i\)
\(L(1)\) \(\approx\) \(0.8790714927 - 0.08875302886i\)
\(L(1)\) \(\approx\) \(0.8790714927 - 0.08875302886i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.984 - 0.173i)T \)
3 \( 1 + (0.893 - 0.448i)T \)
5 \( 1 + (-0.396 + 0.918i)T \)
7 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (0.549 + 0.835i)T \)
13 \( 1 + (-0.116 - 0.993i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.893 - 0.448i)T \)
31 \( 1 + (0.597 - 0.802i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.230 + 0.973i)T \)
53 \( 1 + (-0.802 + 0.597i)T \)
59 \( 1 + (0.998 - 0.0581i)T \)
61 \( 1 + (-0.686 + 0.727i)T \)
67 \( 1 + (0.918 - 0.396i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (0.116 - 0.993i)T \)
83 \( 1 + (-0.893 + 0.448i)T \)
89 \( 1 + (0.686 - 0.727i)T \)
97 \( 1 + (-0.396 + 0.918i)T \)
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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

−18.86143264148223002502576279363, −17.35384111193419858101030361777, −16.94770025037470445572556003924, −16.28357611103103706917224688287, −15.86010039300926311670298352371, −15.24608602146666543434951188221, −14.36774338923660008862302098232, −13.73225016369568745247499965914, −12.81911748470455779576133378808, −12.21188398040467697057789527372, −11.30701831777508928177174207921, −10.56209707074956564981388707223, −9.83486077265910583351557375111, −9.13848209832988335729771971269, −8.69690142440524182888192089507, −8.32696659433890547456666355028, −7.17717542700906375311547206584, −6.76005864850935065927326469538, −5.753935441505178540906343716551, −4.82179952603566958439852937486, −3.85680701154906085844611387160, −3.283910270618588008300111866265, −2.3623500755196253349137008397, −1.37731626462271534484501565581, −0.641160740050005223964585282932, 0.48495142448309226469649313939, 1.296962733151517745209879507956, 2.51664675474724458725506160564, 2.81793542482151970937776987134, 3.48142480195826382802397073434, 4.38027902466288848715987260411, 6.1017526707492818802943535912, 6.442933578942568806564268963255, 7.34301680474726934292998091610, 7.65438890267994297590848568825, 8.43635718432580746682686239480, 9.32963500178557428100532246815, 9.88806708262866125683443140864, 10.29666069166308199326551500144, 11.29380176564469524314289082625, 12.19022430254223396551692476767, 12.52658385823885653643568616898, 13.34350751258608346469127335677, 14.41074750366890994964775563614, 14.91736244126038881370888369460, 15.50127094778823300292410203048, 16.09647370290739780706142771836, 17.0817714713204366267217222708, 17.74256167580812942006045413925, 18.43016569591054418558191963023

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