↑

Properties

Label 1-4033-4033.567-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.430 - 0.902i$
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  + (−0.5 − 0.866i)2-s + (−0.686 − 0.727i)3-s + (−0.5 + 0.866i)4-s + (−0.835 + 0.549i)5-s + (−0.286 + 0.957i)6-s + (−0.0581 − 0.998i)7-s + 8-s + (−0.0581 + 0.998i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (0.973 − 0.230i)12-s + (−0.835 + 0.549i)13-s + (−0.835 + 0.549i)14-s + (0.973 + 0.230i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.686 − 0.727i)3-s + (−0.5 + 0.866i)4-s + (−0.835 + 0.549i)5-s + (−0.286 + 0.957i)6-s + (−0.0581 − 0.998i)7-s + 8-s + (−0.0581 + 0.998i)9-s + (0.893 + 0.448i)10-s + (0.893 − 0.448i)11-s + (0.973 − 0.230i)12-s + (−0.835 + 0.549i)13-s + (−0.835 + 0.549i)14-s + (0.973 + 0.230i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.430 - 0.902i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.430 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4509957889 - 0.2844347347i\)
\(L(\frac12)\) \(\approx\) \(0.4509957889 - 0.2844347347i\)
\(L(1)\) \(\approx\) \(0.4465331504 - 0.2588763635i\)
\(L(1)\) \(\approx\) \(0.4465331504 - 0.2588763635i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.686 - 0.727i)T \)
5 \( 1 + (-0.835 + 0.549i)T \)
7 \( 1 + (-0.0581 - 0.998i)T \)
11 \( 1 + (0.893 - 0.448i)T \)
13 \( 1 + (-0.835 + 0.549i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.835 + 0.549i)T \)
31 \( 1 + (0.597 + 0.802i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.286 - 0.957i)T \)
53 \( 1 + (-0.686 - 0.727i)T \)
59 \( 1 + (0.973 + 0.230i)T \)
61 \( 1 + (-0.993 - 0.116i)T \)
67 \( 1 + (0.973 + 0.230i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.835 - 0.549i)T \)
79 \( 1 + (-0.286 - 0.957i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (-0.835 + 0.549i)T \)
97 \( 1 + (0.597 - 0.802i)T \)
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

−18.53879260422376243600360175788, −17.48471773387056536101580382812, −17.28178263954172293771842451066, −16.4498175218497854159319880900, −16.00711148787891520031801144026, −15.20341408322084017191394071059, −14.91840266463915283787018242697, −14.31787549715437946516132598432, −12.84497897392323986165712103078, −12.40160198215107002043482099190, −11.67739161793112904825129741793, −11.03233755604095426099428311867, −10.00252407282613954549160746340, −9.51086202501335345416101711003, −8.97264444183394687019108106984, −8.069533432626262742197277925463, −7.57481728295958182751043630053, −6.4595562364687522648885090193, −5.940700445538975914009584315431, −5.231226634278991086780721874, −4.44639113354769685833952548966, −4.0547496038524838941619986448, −2.76151975519187054707132705803, −1.48142467357751620690949745393, −0.41657626891390477105699895036, 0.51856196173472801786805863028, 1.40378165796443267108259674131, 2.18518385908598799053035787396, 3.2735960808116054234500051862, 3.94756329971734821620425723184, 4.52324106380893870086949720100, 5.671814326930104959522812830700, 6.80406468145678093274301323174, 7.11610061147225149718354899204, 7.88020428593502166826882639914, 8.43932854463376452634664849560, 9.553077679781386734412108513718, 10.33566337331013818034156583185, 10.81803865642520954661528404923, 11.50539905041540437996955302120, 12.10179404706042345783772676577, 12.474147615841677426563844490, 13.4483116584733003271535281475, 14.206613209447989540768346936916, 14.57316719480825493687833919197, 16.15199296145736018074120153371, 16.452939490868542946592178480510, 17.23416851732562800486416572347, 17.58145537543369086144061358623, 18.62124659012728884491756866193

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