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Properties

Label 1-4033-4033.1028-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.829 - 0.558i$
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  + (−0.939 − 0.342i)2-s + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (−0.727 + 0.686i)5-s + (0.835 − 0.549i)6-s + (0.973 − 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.973 + 0.230i)12-s + (0.973 − 0.230i)13-s + (−0.993 − 0.116i)14-s + (−0.116 − 0.993i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.597 + 0.802i)3-s + (0.766 + 0.642i)4-s + (−0.727 + 0.686i)5-s + (0.835 − 0.549i)6-s + (0.973 − 0.230i)7-s + (−0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.918 − 0.396i)10-s + (−0.918 − 0.396i)11-s + (−0.973 + 0.230i)12-s + (0.973 − 0.230i)13-s + (−0.993 − 0.116i)14-s + (−0.116 − 0.993i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1028, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.829 - 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5755415418 - 0.1758590998i\)
\(L(\frac12)\) \(\approx\) \(0.5755415418 - 0.1758590998i\)
\(L(1)\) \(\approx\) \(0.5401461806 + 0.04790775182i\)
\(L(1)\) \(\approx\) \(0.5401461806 + 0.04790775182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (-0.727 + 0.686i)T \)
7 \( 1 + (0.973 - 0.230i)T \)
11 \( 1 + (-0.918 - 0.396i)T \)
13 \( 1 + (0.973 - 0.230i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.802 - 0.597i)T \)
31 \( 1 + (-0.957 + 0.286i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (-0.448 - 0.893i)T \)
53 \( 1 + (0.957 + 0.286i)T \)
59 \( 1 + (-0.993 + 0.116i)T \)
61 \( 1 + (-0.998 + 0.0581i)T \)
67 \( 1 + (0.727 + 0.686i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.396 + 0.918i)T \)
79 \( 1 + (0.973 + 0.230i)T \)
83 \( 1 + (-0.597 + 0.802i)T \)
89 \( 1 + (0.998 - 0.0581i)T \)
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

−18.38184815779837190029106133329, −17.95996200124428772716822393814, −17.14305386483515623493158117263, −16.78291047985251149366625727206, −15.877141867104717823514093121733, −15.36840136229211368967758754995, −14.74113181283823709396755269968, −13.620100349612909851442139102826, −12.95066674782816628858909358286, −12.2703118333241674999993360468, −11.39933238142946325062224385666, −11.04757685162778786672479997404, −10.54661289695404043085415677065, −9.1348681268544654156746565744, −8.72713195502670468648027855175, −7.946146265838328502996042278366, −7.53768312630727129538406206512, −6.84675421370505373772016391835, −5.86908549580559985733779074988, −5.21418846053372858267016923285, −4.693122397548142498875641193576, −3.35540459469152684256876022132, −2.06806566561887326322113023358, −1.601322442308915998432310089124, −0.73767564181526933053858296183, 0.37620010088101063218397323698, 1.33743028804088823278503182373, 2.54947121803363997445145431246, 3.36025340722820184684757915295, 3.88681665992600251923586125280, 4.89221629500739916989836625874, 5.692865356846397862891138583121, 6.601335262921446433791902211462, 7.38408433063955058660635385214, 8.065477422734283713892208620096, 8.62187014175487265988491439696, 9.51827443222215418093453237421, 10.418950734037755370695660877866, 10.748570057365571563899802333355, 11.40909901077731540095769997559, 11.66318645098801101551964434803, 12.66956136117853160565905973894, 13.62468712060466153653588596703, 14.664975617055462635438385039272, 15.22977590963028563085485200674, 15.80264084417386500842049141552, 16.513967363946558107972462927056, 16.906155557589410782405530460920, 18.0477205483197290042782353491, 18.36079000408042645439614233329

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