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Properties

Label 1-4033-4033.52-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.770 - 0.636i$
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  + (0.766 − 0.642i)2-s + (−0.973 − 0.230i)3-s + (0.173 − 0.984i)4-s + (−0.448 − 0.893i)5-s + (−0.893 + 0.448i)6-s + (−0.835 − 0.549i)7-s + (−0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.918 − 0.396i)10-s + (0.918 − 0.396i)11-s + (−0.396 + 0.918i)12-s + (−0.0581 + 0.998i)13-s + (−0.993 + 0.116i)14-s + (0.230 + 0.973i)15-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.973 − 0.230i)3-s + (0.173 − 0.984i)4-s + (−0.448 − 0.893i)5-s + (−0.893 + 0.448i)6-s + (−0.835 − 0.549i)7-s + (−0.5 − 0.866i)8-s + (0.893 + 0.448i)9-s + (−0.918 − 0.396i)10-s + (0.918 − 0.396i)11-s + (−0.396 + 0.918i)12-s + (−0.0581 + 0.998i)13-s + (−0.993 + 0.116i)14-s + (0.230 + 0.973i)15-s + (−0.939 − 0.342i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.770 - 0.636i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.770 - 0.636i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8076966503 - 0.2904992334i\)
\(L(\frac12)\) \(\approx\) \(0.8076966503 - 0.2904992334i\)
\(L(1)\) \(\approx\) \(0.7122380960 - 0.5271833607i\)
\(L(1)\) \(\approx\) \(0.7122380960 - 0.5271833607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-0.973 - 0.230i)T \)
5 \( 1 + (-0.448 - 0.893i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (0.918 - 0.396i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.918 - 0.396i)T \)
31 \( 1 + (-0.230 - 0.973i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.549 + 0.835i)T \)
53 \( 1 + (-0.918 - 0.396i)T \)
59 \( 1 + (0.973 - 0.230i)T \)
61 \( 1 + (0.116 + 0.993i)T \)
67 \( 1 + (-0.116 + 0.993i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.597 + 0.802i)T \)
79 \( 1 + (0.597 - 0.802i)T \)
83 \( 1 + (0.835 + 0.549i)T \)
89 \( 1 + (-0.727 - 0.686i)T \)
97 \( 1 + (0.727 + 0.686i)T \)
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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

−18.20226730001537566124183577885, −17.72908399864799291578222919151, −17.19529833528129957276331276688, −16.276391283013301557392089949805, −15.61031257310533966236920637536, −15.39983700143653127730058923353, −14.69675844134373241293056249004, −13.7943896211787459222789606093, −12.95601768936010743922894855004, −12.414411676718584791371676094797, −11.778782400094330019191030188497, −11.143625871111398572176665374177, −10.52419457896338127311428090051, −9.440489772034561577533187435496, −8.92424095303270428612349864316, −7.64603841481730530257739796298, −6.994462105510194893272210167198, −6.637851641556528743554801233360, −5.83739926017627472962693643215, −5.262770414131452437343676681881, −4.31797372270046871887943452701, −3.65726484094025007373195374340, −2.98676635202682238421237513335, −2.04209732106773616054825495303, −0.28990209180591505699628876273, 0.74360810744910362262509776276, 1.49011218167177848160403363804, 2.32399158601698107488636705268, 3.80738939548499904977135448471, 4.12443122347324510999417885064, 4.58596273177149553452086625783, 5.83593138545384882707729638345, 6.165769624717159511948349260594, 6.83791874011740604581076486621, 7.75731181353020100497618035837, 8.95798262251515808593974155345, 9.50099467147203713277194531534, 10.3351736155345355211209724769, 11.087514662099940821479672937379, 11.61239493781367568344407630249, 12.24243241451016845708519266624, 12.9559480095526850021359351941, 13.16997634869362554850150297924, 14.17554266394340540700688061188, 14.876413981538654614358190297105, 15.90591679279593641960960631530, 16.37906900533662961675005744041, 16.84105731110380700606359297086, 17.566849568680065750646164325364, 18.81107394960075728691147271445

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