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Properties

Label 1-280-280.109-r0-0-0
Degree $1$
Conductor $280$
Sign $0.832 - 0.553i$
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  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + 13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + 27-s − 29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)39-s + 41-s + 43-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + 13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + 27-s − 29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)39-s + 41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(1.30031\)
Root analytic conductor: \(1.30031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 280,\ (0:\ ),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.057695125 - 0.3196505528i\)
\(L(\frac12)\) \(\approx\) \(1.057695125 - 0.3196505528i\)
\(L(1)\) \(\approx\) \(0.9620467282 - 0.2015186408i\)
\(L(1)\) \(\approx\) \(0.9620467282 - 0.2015186408i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 - T \)
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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

−25.78102678611057588264213983678, −24.87008562240857385972527009391, −23.69021049142284162078918394407, −22.88881150639989358692042264629, −22.13793800414478497290296039827, −21.101640579194249941863882119588, −20.60209083026203080347787260981, −19.30422215080468834925139239670, −18.30710649336368615254756846868, −17.330907177629300123616907385861, −16.26537359401374323037506557002, −15.90872451920401892148551963848, −14.57021231262664914579193170101, −13.82248484837155931397616938077, −12.42931800249368781203993710400, −11.37685622678466504099049623807, −10.79464067376128431374642883848, −9.5212685469525219388600388998, −8.8443434710683687459249664838, −7.43604367018889206219703674117, −6.0187680219113764472878781228, −5.39725640925408532313560054985, −3.95113081723835172659608851188, −3.20882201459954928920062071353, −1.12421128371907955955025867058, 1.10342691245635058258519547359, 2.282788293285251364703103887830, 3.868940935831639178445722473391, 5.22165329370829060035337241835, 6.27886208013483323463123356918, 7.14129459075250612364891834198, 8.17767216162773610660704884524, 9.31594145619724311020939487185, 10.67493415888036075366226940900, 11.483054410157002461069546390486, 12.531801319718843166225908031300, 13.20811380012794691010411417241, 14.300455271384991576089511132433, 15.34421300039859640475239929163, 16.589748139028250629030519899292, 17.327468829465441477081933186582, 18.23256430781550211806396178796, 19.00072266650999861189042479803, 19.98786506663765420813483378277, 20.9081055839297387125571771131, 22.25231277372206953004960874831, 22.81671005513546996717320906810, 23.79831413808585003013193600678, 24.477374100278974991011299192915, 25.54421630348109465000245383515

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