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Properties

Label 1-4033-4033.248-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.957 - 0.287i$
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 + (−0.0581 − 0.998i)3-s + (0.939 − 0.342i)4-s + (0.597 − 0.802i)5-s + (−0.230 − 0.973i)6-s + (0.396 + 0.918i)7-s + (0.866 − 0.5i)8-s + (−0.993 + 0.116i)9-s + (0.448 − 0.893i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)12-s + (−0.918 + 0.396i)13-s + (0.549 + 0.835i)14-s + (−0.835 − 0.549i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (−0.0581 − 0.998i)3-s + (0.939 − 0.342i)4-s + (0.597 − 0.802i)5-s + (−0.230 − 0.973i)6-s + (0.396 + 0.918i)7-s + (0.866 − 0.5i)8-s + (−0.993 + 0.116i)9-s + (0.448 − 0.893i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)12-s + (−0.918 + 0.396i)13-s + (0.549 + 0.835i)14-s + (−0.835 − 0.549i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.091167299 - 0.7477109237i\)
\(L(\frac12)\) \(\approx\) \(5.091167299 - 0.7477109237i\)
\(L(1)\) \(\approx\) \(2.025380376 - 0.7182146499i\)
\(L(1)\) \(\approx\) \(2.025380376 - 0.7182146499i\)

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 \( 1 + (-0.0581 - 0.998i)T \)
5 \( 1 + (0.597 - 0.802i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (0.448 + 0.893i)T \)
13 \( 1 + (-0.918 + 0.396i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (0.0581 + 0.998i)T \)
31 \( 1 + (0.993 - 0.116i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.727 + 0.686i)T \)
53 \( 1 + (-0.116 + 0.993i)T \)
59 \( 1 + (0.549 - 0.835i)T \)
61 \( 1 + (0.286 - 0.957i)T \)
67 \( 1 + (-0.802 + 0.597i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.893 - 0.448i)T \)
79 \( 1 + (0.918 + 0.396i)T \)
83 \( 1 + (0.0581 + 0.998i)T \)
89 \( 1 + (-0.286 + 0.957i)T \)
97 \( 1 + (0.597 - 0.802i)T \)
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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

−18.00583708292341449752104935919, −17.34133030394558109058196126228, −16.96769239564826817338818777561, −16.200169197545834679010241248812, −15.3750693120676612721808378906, −14.89153667527772124368845194631, −14.28793983105870940399354696988, −13.600552577948898697719818226236, −13.35748320387210789394106659942, −11.83420911270559883994533950670, −11.57554198257211896092094807387, −10.674316118797689251700127667686, −10.318183527407072570110821877420, −9.60443126770763546125350364399, −8.46341967906672938099256433988, −7.75074205184902669934393239898, −6.84913951004901510321071484383, −6.21482183166820098946175700811, −5.54475512415794168881446879146, −4.75404113639297171931676062489, −4.079298453113340489069775718559, −3.3936287481927453681747944973, −2.71731065381073500843407217046, −1.86465543436738557981114217244, −0.50254301583249369851959332532, 0.90887305281236493679495666011, 1.77602775416385824507564729475, 2.16930930848349936387271272017, 2.87409455372600693912331237808, 4.19344431943001554171314689824, 4.9597327360495033220008175921, 5.403515301622588746337176688049, 6.21921914876483786790191742708, 6.835029592922541482859659213632, 7.630069829493309494432656421390, 8.39205929595259928317708985545, 9.35340205182902334620900744005, 9.86375342535484931086375654451, 11.02971900348554926628683297403, 11.967182961127059799022072907139, 12.15163094723788836722791541824, 12.58621550821484220185535268938, 13.49837913787513812377158490226, 14.16757501351057748392143036472, 14.46904572394317288605905474946, 15.44769958042099445077471177094, 16.1727722240035832836799863545, 16.94535094096032273575953728934, 17.63679637154843304943833758927, 18.214927179869272735721243231432

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