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Properties

Label 1-4033-4033.215-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.767 + 0.640i$
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 + (0.597 + 0.802i)3-s + (−0.5 − 0.866i)4-s + (−0.973 − 0.230i)5-s + (−0.993 + 0.116i)6-s + (−0.286 − 0.957i)7-s + 8-s + (−0.286 + 0.957i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (0.396 − 0.918i)12-s + (0.973 + 0.230i)13-s + (0.973 + 0.230i)14-s + (−0.396 − 0.918i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.597 + 0.802i)3-s + (−0.5 − 0.866i)4-s + (−0.973 − 0.230i)5-s + (−0.993 + 0.116i)6-s + (−0.286 − 0.957i)7-s + 8-s + (−0.286 + 0.957i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (0.396 − 0.918i)12-s + (0.973 + 0.230i)13-s + (0.973 + 0.230i)14-s + (−0.396 − 0.918i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.767 + 0.640i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.767 + 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4710385148 + 1.299887898i\)
\(L(\frac12)\) \(\approx\) \(0.4710385148 + 1.299887898i\)
\(L(1)\) \(\approx\) \(0.7224238562 + 0.5651780774i\)
\(L(1)\) \(\approx\) \(0.7224238562 + 0.5651780774i\)

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 \( 1 + (0.597 + 0.802i)T \)
5 \( 1 + (-0.973 - 0.230i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (0.686 + 0.727i)T \)
13 \( 1 + (0.973 + 0.230i)T \)
17 \( 1 + T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.973 - 0.230i)T \)
31 \( 1 + (0.0581 + 0.998i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.993 + 0.116i)T \)
53 \( 1 + (-0.597 - 0.802i)T \)
59 \( 1 + (0.396 + 0.918i)T \)
61 \( 1 + (0.835 + 0.549i)T \)
67 \( 1 + (-0.396 - 0.918i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.973 - 0.230i)T \)
79 \( 1 + (-0.993 - 0.116i)T \)
83 \( 1 + (0.893 - 0.448i)T \)
89 \( 1 + (-0.973 - 0.230i)T \)
97 \( 1 + (0.0581 - 0.998i)T \)
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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

−18.57008030720900708096311094899, −17.89576964702685758064334780450, −16.978142156074767826402536205487, −16.19506243695929195706245788961, −15.53646365477652380692669786365, −14.63554050421189667362683500766, −14.08090056680687032251044611888, −13.099752110833058448426003996773, −12.66420383134585981144479739504, −12.006286456371552385066726345236, −11.31811915592992736021932429625, −10.9791921839421337203683574281, −9.68679387450549855811042569592, −9.03557917517334706051316683266, −8.54397616529484843657675218024, −7.96026468315328824230288238236, −7.24105058484306683625980449734, −6.39118543667262366549651656017, −5.5768843100679495537654994801, −4.20801462449828615613200494695, −3.59410450092136885499824932769, −2.9218622415563285022316790477, −2.399687901767439610990110348453, −1.18813295994889516053165098553, −0.58534929799751700184028972226, 0.970719737353581783907712725291, 1.71179406501852894974156454967, 3.38037109520666732912066258818, 3.859496228886808541103996709039, 4.353038416737418233496110686282, 5.28967910400623537045894722882, 6.08253451907106212823538269964, 7.2524569443252468278709458423, 7.50621200765315185487227495416, 8.23078764248945725362820929009, 9.05077688827130774389273807737, 9.52194138077961287294975856959, 10.33059027863340331622177979061, 10.85808066351386783680594514142, 11.71757722565737813655607566602, 12.73419560118709715385455277480, 13.58651794452563458096972186459, 14.29859646150234901256757000537, 14.663840576359982215933127584686, 15.56843816692742912823583316883, 15.97000887060164254027407936549, 16.63677617287700954847006433367, 17.00109653925634654057658432163, 17.94175968518477397350297808377, 18.95518135985571126911234234278

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