↑

Properties

Label 1-4029-4029.236-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.0758 + 0.997i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  − i·2-s − 4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)7-s + i·8-s + (0.707 + 0.707i)10-s + (0.707 + 0.707i)11-s − 13-s + (−0.707 + 0.707i)14-s + 16-s + i·19-s + (0.707 − 0.707i)20-s + (0.707 − 0.707i)22-s + (0.707 + 0.707i)23-s − i·25-s + i·26-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.707 + 0.707i)5-s + (−0.707 − 0.707i)7-s + i·8-s + (0.707 + 0.707i)10-s + (0.707 + 0.707i)11-s − 13-s + (−0.707 + 0.707i)14-s + 16-s + i·19-s + (0.707 − 0.707i)20-s + (0.707 − 0.707i)22-s + (0.707 + 0.707i)23-s − i·25-s + i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.0758 + 0.997i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.0758 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3758359954 + 0.3483196782i\)
\(L(\frac12)\) \(\approx\) \(0.3758359954 + 0.3483196782i\)
\(L(1)\) \(\approx\) \(0.6831298493 - 0.1770450926i\)
\(L(1)\) \(\approx\) \(0.6831298493 - 0.1770450926i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 - iT \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 - T \)
19 \( 1 + iT \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + iT \)
47 \( 1 - T \)
53 \( 1 + iT \)
59 \( 1 - iT \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + T \)
97 \( 1 + (0.707 - 0.707i)T \)
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

−18.29434231452945055264501042012, −17.36302229762231508819199453695, −16.76019161118403400687885660443, −16.311370997153224082803166347876, −15.67002947668131542268729747015, −14.97328833774254000024585778674, −14.52831372566715419051936311519, −13.465402707638431087889618509441, −12.942588869760606259897183619791, −12.26339768983053681881883579870, −11.68883621895076494762137053417, −10.67098230752886315424754242665, −9.57466302118661318486691871815, −9.101363368354571120115780517963, −8.61974002610644049395920863915, −7.84547332240984563902860239295, −6.96197110321203707502630707141, −6.511922009157951336495982934368, −5.506770556271299373900172379290, −4.99063900586087718531692675468, −4.198712693203890360077517170538, −3.40784438022878744528139434721, −2.54327743894316199639315535594, −1.02902351778336072336377965179, −0.19864559940431652937697237944, 0.99002157476927364547018733265, 1.961960376278080528939607680946, 2.93800784099834657638088333792, 3.45220201400406496851106120508, 4.23949396505846401167157661, 4.747131201404081930192046107901, 5.97123741703570532498236188958, 6.80742552791997971055039450894, 7.54584370293126182984899556157, 8.08627864434849638290130929366, 9.38163366264965544955658223666, 9.664354784126728975199929961762, 10.40954624835701464402145498211, 11.07312673622763818761251234480, 11.74885913283327768914300279952, 12.453848743813401309483406653308, 12.868758949842400439954588923860, 13.89850503081295407336924439979, 14.501404486526985917907523973959, 14.921636261004494096842380470321, 15.97326889849234004502557932349, 16.7376239565616830711234050784, 17.44044320693139577478768934437, 18.02058563627377657176250339867, 18.95516320779474097496516736

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