Properties

Label 1-4033-4033.1844-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.0513 + 0.998i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0513 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0513 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0513 + 0.998i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.0513 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3672473244 - 0.3488490150i\)
\(L(\frac12)\) \(\approx\) \(-0.3672473244 - 0.3488490150i\)
\(L(1)\) \(\approx\) \(0.4675246717 - 0.4108707524i\)
\(L(1)\) \(\approx\) \(0.4675246717 - 0.4108707524i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.642 - 0.766i)T \)
3 \( 1 + (-0.893 - 0.448i)T \)
5 \( 1 + (0.918 - 0.396i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (0.893 - 0.448i)T \)
13 \( 1 + (0.116 - 0.993i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (0.998 - 0.0581i)T \)
31 \( 1 + (-0.116 - 0.993i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.686 - 0.727i)T \)
53 \( 1 + (-0.396 - 0.918i)T \)
59 \( 1 + (-0.998 - 0.0581i)T \)
61 \( 1 + (0.727 - 0.686i)T \)
67 \( 1 + (-0.993 + 0.116i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.893 + 0.448i)T \)
79 \( 1 + (-0.802 + 0.597i)T \)
83 \( 1 + (-0.835 + 0.549i)T \)
89 \( 1 + (0.230 + 0.973i)T \)
97 \( 1 + (-0.802 - 0.597i)T \)
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

−18.80987155841123277879505752389, −17.86355933336237076504751776168, −17.2474386504406298218948669188, −16.92324570881960135611481112443, −16.32241200889567025854053970902, −15.5446465240138000436455459514, −14.85412669010285917781465018833, −14.33423650098769277954580450070, −13.44209761737182150040687926417, −12.64302578511974547530998288931, −11.92141558616132805121425085449, −10.826305730064975343839535876489, −10.46068902212784758510753513382, −9.80856332961283712171619824559, −9.071026612387958251600228891000, −8.81297906788398312921527495165, −7.2492845676908121438527348586, −6.615124190451622814080932839400, −6.375502208121637276757350228540, −5.7299670267707245070914419001, −4.76261907189333091131233066339, −4.12211428377818544871011123942, −2.98471169877159442640834224036, −1.7302654436589805542506589502, −1.161170681661045486091190678851, 0.14268939944611561308075786200, 0.77800695401671572972434980016, 1.40264305288238995916493994833, 2.4328690857531904655063570027, 3.133476936125938946257977737241, 4.128641853073264204744764067205, 5.05447674875319732634425690174, 5.840102433318605826700183552454, 6.603967161250127927570611835000, 7.099284784054456234704129024281, 8.21305627408400506614189760429, 8.84312149051969287697959477154, 9.78845262599380569124931052555, 10.03584413291844144487735059162, 10.883348948628973799020823473677, 11.54038446345940709328295413900, 12.37422469946984043748784663315, 12.78803166663678240497990807092, 13.42780742744989457138652175780, 13.91094810237436941999017794976, 15.26172007419342869759942385683, 16.23751708932232497020170048295, 16.70733474023932641591962868231, 17.1810811422245448049663032764, 17.70434878051796208621869719048

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