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Properties

Label 1-4033-4033.2191-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.805 - 0.593i$
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 + (0.686 + 0.727i)3-s + (0.173 − 0.984i)4-s + (−0.448 + 0.893i)5-s + (0.993 + 0.116i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.230 + 0.973i)10-s + (−0.230 + 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.835 + 0.549i)13-s + (−0.286 + 0.957i)14-s + (−0.957 + 0.286i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.686 + 0.727i)3-s + (0.173 − 0.984i)4-s + (−0.448 + 0.893i)5-s + (0.993 + 0.116i)6-s + (−0.835 + 0.549i)7-s + (−0.5 − 0.866i)8-s + (−0.0581 + 0.998i)9-s + (0.230 + 0.973i)10-s + (−0.230 + 0.973i)11-s + (0.835 − 0.549i)12-s + (−0.835 + 0.549i)13-s + (−0.286 + 0.957i)14-s + (−0.957 + 0.286i)15-s + (−0.939 − 0.342i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.805 - 0.593i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.805 - 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01964892046 + 0.05980302524i\)
\(L(\frac12)\) \(\approx\) \(0.01964892046 + 0.05980302524i\)
\(L(1)\) \(\approx\) \(1.222401995 + 0.1482883203i\)
\(L(1)\) \(\approx\) \(1.222401995 + 0.1482883203i\)

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 \( 1 + (0.686 + 0.727i)T \)
5 \( 1 + (-0.448 + 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.230 + 0.973i)T \)
13 \( 1 + (-0.835 + 0.549i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.727 - 0.686i)T \)
31 \( 1 + (-0.998 - 0.0581i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.918 + 0.396i)T \)
53 \( 1 + (0.998 - 0.0581i)T \)
59 \( 1 + (-0.286 - 0.957i)T \)
61 \( 1 + (0.802 + 0.597i)T \)
67 \( 1 + (0.448 + 0.893i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.973 - 0.230i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (0.686 + 0.727i)T \)
89 \( 1 + (-0.802 - 0.597i)T \)
97 \( 1 + (-0.448 + 0.893i)T \)
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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

−18.00847722496712075827490956933, −17.03343148980960548104474210781, −16.67925305254273619825838413464, −15.97993581378720135361453440700, −15.33128473722140280889697592380, −14.52320855364198225728353580128, −13.914086354624448715821482380429, −13.2730859392017044502060438283, −12.74756778550014061139719084890, −12.245989234391762274972940421593, −11.62888129657923276879003589196, −10.42114822365524116957848960112, −9.52438270467919250577172850655, −8.64982600544923094105282256809, −8.20471242452655798404314757657, −7.4326387661035901458568349188, −7.03098379088465929239171208365, −5.93035650961074804864320111819, −5.52816972144957546577208322285, −4.43668323802162047356654791581, −3.60974317136608740221308488120, −3.22512040452132136193711906388, −2.281243815408102643895433766583, −1.05717506119929033269374933601, −0.01184623369158887851513522625, 1.96347716572870623783620201479, 2.45827489394230307058632586311, 3.00179970693154031281890298010, 3.971510612545473238297688402137, 4.28020023947037644776609124926, 5.26030868887518745861968198497, 6.10435415832863806809433817270, 6.9044623568748205368833158066, 7.5950088176303684811867424657, 8.652308952342854527529941239773, 9.484562859658124502676416592, 10.176999593978673198252996449494, 10.33271436725518597806041014218, 11.357008328267746475294231396452, 12.159114298748933866864370116, 12.59833537236673508771503034676, 13.4650625336718121253852388109, 14.35863568080974060880945953526, 14.71938255452506180568923291359, 15.21962170532660463994728855041, 15.87099566259826687629985582976, 16.500626671684203741077845071489, 17.637488231081546263829450153351, 18.67715956977850304167055885611, 19.08096904874586171041838182613

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