↑

Properties

Label 1-4033-4033.138-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.972 - 0.231i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.286 − 0.957i)3-s + (0.173 + 0.984i)4-s + (0.0581 − 0.998i)5-s + (0.396 − 0.918i)6-s + (0.893 − 0.448i)7-s + (−0.5 + 0.866i)8-s + (−0.835 + 0.549i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (0.893 − 0.448i)12-s + (0.893 − 0.448i)13-s + (0.973 + 0.230i)14-s + (−0.973 + 0.230i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.286 − 0.957i)3-s + (0.173 + 0.984i)4-s + (0.0581 − 0.998i)5-s + (0.396 − 0.918i)6-s + (0.893 − 0.448i)7-s + (−0.5 + 0.866i)8-s + (−0.835 + 0.549i)9-s + (0.686 − 0.727i)10-s + (0.686 + 0.727i)11-s + (0.893 − 0.448i)12-s + (0.893 − 0.448i)13-s + (0.973 + 0.230i)14-s + (−0.973 + 0.230i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.972 - 0.231i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.972 - 0.231i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.850393857 - 0.3341673991i\)
\(L(\frac12)\) \(\approx\) \(2.850393857 - 0.3341673991i\)
\(L(1)\) \(\approx\) \(1.655618560 - 0.03238775248i\)
\(L(1)\) \(\approx\) \(1.655618560 - 0.03238775248i\)

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 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (0.0581 - 0.998i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (0.686 + 0.727i)T \)
13 \( 1 + (0.893 - 0.448i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.286 + 0.957i)T \)
31 \( 1 + (0.835 - 0.549i)T \)
41 \( 1 - T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.597 + 0.802i)T \)
53 \( 1 + (0.835 + 0.549i)T \)
59 \( 1 + (0.973 - 0.230i)T \)
61 \( 1 + (0.993 - 0.116i)T \)
67 \( 1 + (0.0581 + 0.998i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.686 - 0.727i)T \)
79 \( 1 + (0.893 + 0.448i)T \)
83 \( 1 + (-0.286 - 0.957i)T \)
89 \( 1 + (0.993 - 0.116i)T \)
97 \( 1 + (0.0581 - 0.998i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.640666077290842694628505017489, −17.79177111649954362602550507104, −17.290386272040599924717387452037, −16.17594284641741708645852964125, −15.486205219104455852584660556749, −15.06453956045555072550835705796, −14.43173024997872779307742998729, −13.81017463094236178413132447406, −13.24482611429086752860415339686, −11.79506300166972163958630319372, −11.61614222330672243120416407358, −11.010220166785754746881208088543, −10.58196905358657918417356198241, −9.676591152670220367968795262023, −8.91959926971653018239669635178, −8.35244038164913765808908863112, −6.77795846620159085445469755034, −6.42602432982925255223874661585, −5.632063382770848296432366552571, −4.89815221565095144294274471490, −4.097348466860879871044524387174, −3.60840583641171214696980886781, −2.70184438027040598150338567633, −2.04597619562695759203377863638, −0.869855663608052963234024527095, 0.813333967001408664423455391479, 1.68715499919743792532212085989, 2.34851373383153622868916498756, 3.669384144368808144668952955950, 4.48062848985119354140391112295, 4.897026805776825392273816543684, 5.814594875183532431889509491855, 6.46288258944814369159970983186, 7.09808800362021411763604418344, 7.90792859155767875592644833178, 8.613835209963332837460601595502, 8.80606309131898954875599900296, 10.39237756400636574149568933522, 11.21057837390586468866283935748, 11.77302816445193454779968945220, 12.58379485316031602668557227826, 12.95219412209064668072699326182, 13.56774489499008560001802593805, 14.28048761710047793628197650940, 14.9286987733007699633181470060, 15.6773321228073071952057202194, 16.6412164580097152283530502994, 17.02803022957226692716204246331, 17.65086901092814910968451579162, 17.99883244996407809124139681820

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