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Properties

Label 1-455-455.279-r0-0-0
Degree $1$
Conductor $455$
Sign $-0.884 - 0.466i$
Analytic cond. $2.11301$
Root an. cond. $2.11301$
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  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)6-s − i·8-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s − 12-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + i·18-s + (−0.866 − 0.5i)19-s + (0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.866 + 0.5i)24-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)6-s − i·8-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s − 12-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + i·18-s + (−0.866 − 0.5i)19-s + (0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.866 + 0.5i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(455\)    =    \(5 \cdot 7 \cdot 13\)
Sign: $-0.884 - 0.466i$
Analytic conductor: \(2.11301\)
Root analytic conductor: \(2.11301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{455} (279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 455,\ (0:\ ),\ -0.884 - 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4177396613 - 1.688791611i\)
\(L(\frac12)\) \(\approx\) \(0.4177396613 - 1.688791611i\)
\(L(1)\) \(\approx\) \(1.030090221 - 0.9896926335i\)
\(L(1)\) \(\approx\) \(1.030090221 - 0.9896926335i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + iT \)
53 \( 1 - T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 - iT \)
89 \( 1 + (-0.866 + 0.5i)T \)
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

−24.017219339675659377900401571507, −23.40265082771634362418881538784, −22.54926185890907529654833204257, −21.93275296358830171644912674458, −21.17361926879509026758688827859, −20.40170901992694441561015730620, −19.42106287583365612670032684639, −17.88480910457944217970214760601, −17.04959329040750163262714589109, −16.57465387853126929175264758741, −15.50175066572938230411087008571, −14.84110899822588105948073175334, −14.18286480966261679199436829512, −12.85016625812466751635010655791, −12.099642613182512534738248023058, −11.271090459458382607130838828388, −10.27704480988888372062451909078, −9.18352343600341741108369400319, −8.11487289393498617414526281174, −6.87129122649781065650183175710, −5.998285281449471474339231121864, −5.20088517371264499100585671464, −4.0230437286648571026697970161, −3.60382660848116239904155773705, −1.91723834177873102292230827443, 0.7822118799751810795713917266, 1.95943447939341365606275860646, 3.02213776837140402749354312563, 4.317201275759135750710779701365, 5.33997019065358607450644745285, 6.3235727829709097369599916374, 6.92427144779949573942778364554, 8.232432650894667067466322695128, 9.521699755335563369171205605892, 10.73935060712189096839519770368, 11.44797191749896524841771255556, 12.22740811684042847064311884168, 12.94963738724127478209827559485, 13.96451389025727841258779041670, 14.43725636311220108418163966324, 15.78709655807313243016510527067, 16.65784435096976446132280257009, 17.643886672167084245080998241182, 18.771378919497459462214620138262, 19.272448832327237661493401672622, 20.15908177506669636985798906227, 21.15534071338389589237733909866, 22.12086335117055080728757603905, 22.72601434331107966699921578986, 23.478866741100227566390805540570

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