↑

Properties

Label 1-1037-1037.47-r0-0-0
Degree $1$
Conductor $1037$
Sign $0.394 - 0.918i$
Analytic cond. $4.81580$
Root an. cond. $4.81580$
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 + i·3-s + (−0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − 8-s − 9-s + (−0.866 + 0.5i)10-s − i·11-s + (0.866 − 0.5i)12-s + (−0.5 + 0.866i)13-s + (0.866 − 0.5i)14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + i·3-s + (−0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − 8-s − 9-s + (−0.866 + 0.5i)10-s − i·11-s + (0.866 − 0.5i)12-s + (−0.5 + 0.866i)13-s + (0.866 − 0.5i)14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $0.394 - 0.918i$
Analytic conductor: \(4.81580\)
Root analytic conductor: \(4.81580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (0:\ ),\ 0.394 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220058371 - 0.8038446223i\)
\(L(\frac12)\) \(\approx\) \(1.220058371 - 0.8038446223i\)
\(L(1)\) \(\approx\) \(1.085657179 - 0.3523988159i\)
\(L(1)\) \(\approx\) \(1.085657179 - 0.3523988159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + iT \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 - iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + iT \)
41 \( 1 - iT \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (0.866 - 0.5i)T \)
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

−22.03308856258335939928883045326, −20.951564843561153037005167940567, −19.9226598708758214648205228364, −19.53451163123099685801746047891, −18.1306045919555446653332291709, −17.80779763938117172186359151339, −17.278241772121813748037725012525, −16.05823152978936083619624896081, −15.2580574551017359472743008668, −14.5972712775888740007249408057, −13.990890747539391707498070962977, −13.05611638750537848500929880098, −12.29978766227955207411398989698, −11.653061853221944161560683659877, −10.78866935038911446474655228334, −9.41796330603667735782993996413, −8.16317661774115757851623032071, −7.73832020975788732228072934900, −7.17128159367199871134642312380, −6.4458832295502323955394530388, −5.16572945339586920538829621282, −4.57900397738794367887690321451, −3.3451987991522100995846368599, −2.51989283097751251145316774755, −0.96412980691453280228428258655, 0.677954428998105080902093650841, 2.099137224690852934299106459, 3.15029476638421406830407053038, 4.01570598274091459642581198222, 4.70524091747061807768919610118, 5.32552123153530188561044709800, 6.34394575060866882323751397704, 8.10618730425429905060316666729, 8.60126065225305445745903697330, 9.43626207639007350120212484001, 10.39812775502697587758441439366, 11.20021149201700948545399549805, 11.85892109017053125822279017358, 12.22534229228137184414540719489, 13.7038750587553613280886368296, 14.294600229331925972277738892031, 15.04572523795672391999364966918, 15.78413977777988639800153907712, 16.576919064771897745031304838126, 17.45360458688479814330410432201, 18.77795975559044046095634357306, 19.10331559631226631998504864429, 20.16069906360924763408416166906, 20.806733244099770206810757241820, 21.24045057246613204033016585047

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