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Properties

Label 1-1037-1037.684-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

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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: $\chi_{1037} (684, \cdot )$
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 \( 1 + (0.866 + 0.5i)T \)
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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

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

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