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Properties

Label 1-280-280.237-r0-0-0
Degree $1$
Conductor $280$
Sign $-0.850 - 0.525i$
Analytic cond. $1.30031$
Root an. cond. $1.30031$
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  − i·3-s − 9-s − 11-s − i·13-s − i·17-s − 19-s − i·23-s + i·27-s + 29-s − 31-s + i·33-s − i·37-s − 39-s − 41-s + i·43-s + ⋯
L(s)  = 1  − i·3-s − 9-s − 11-s − i·13-s − i·17-s − 19-s − i·23-s + i·27-s + 29-s − 31-s + i·33-s − i·37-s − 39-s − 41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(1.30031\)
Root analytic conductor: \(1.30031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 280,\ (0:\ ),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2119248313 - 0.7460065638i\)
\(L(\frac12)\) \(\approx\) \(0.2119248313 - 0.7460065638i\)
\(L(1)\) \(\approx\) \(0.7310321017 - 0.4291691083i\)
\(L(1)\) \(\approx\) \(0.7310321017 - 0.4291691083i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + T \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 - iT \)
97 \( 1 \)
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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

−25.91251255112925164735362344717, −25.53865611490411895111217631769, −23.795964350722537935038224962047, −23.48705286464729967384031320189, −22.10023200413270466811935558573, −21.465732258361729130112988595853, −20.784358164120280155508488649811, −19.68388517949084654860949764591, −18.81097180949530745737508785752, −17.52112489045325529543050801143, −16.74569715276242389372086911845, −15.78315818245783855411550499644, −15.05846278490723715067708851177, −14.07051006676806837634422614201, −12.99494321970905721799553365718, −11.75324232713093877080773745178, −10.769709316915965719463316491896, −10.02985687722882516852445745773, −8.92246738564111067587495948853, −8.05860682803958950564731656615, −6.57194810765621416666772081169, −5.426763900963508742341262670086, −4.42853868114193710873157287218, −3.399399097028566915294983903702, −2.03581593128225648295897628632, 0.49088247262677169828763705864, 2.17747964398493723526378761173, 3.08986811493675878487047571915, 4.87104117468717572791678125954, 5.89453220507163906171798858202, 7.004551600122389793218751989856, 7.932372597567512161012462408501, 8.76185637204755435709388864606, 10.25807125189456324809441396843, 11.147823851812636631862418636939, 12.403314396118806357089035552050, 12.96873604755715990433382676457, 13.94442604505052441860066020525, 14.96014254606197239784881454290, 16.06401782027712610718775949972, 17.15518678466759927736234721722, 18.15382737591074449084834658828, 18.606360971608364910584058858322, 19.818205936569466664752692497363, 20.48603850021786364528891269692, 21.62616484253732985517936547262, 22.92783970082532274355916195060, 23.31747846733615538194159487999, 24.437629839024303682839378979286, 25.14384878327267493732668012598

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