↑

Properties

Label 1-455-455.342-r0-0-0
Degree $1$
Conductor $455$
Sign $0.999 + 0.0427i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s − i·12-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + i·18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s − i·12-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + i·18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(455\)    =    \(5 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0427i$
Analytic conductor: \(2.11301\)
Root analytic conductor: \(2.11301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{455} (342, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 455,\ (0:\ ),\ 0.999 + 0.0427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.301815341 + 0.02786261935i\)
\(L(\frac12)\) \(\approx\) \(1.301815341 + 0.02786261935i\)
\(L(1)\) \(\approx\) \(1.038194967 + 0.03880145379i\)
\(L(1)\) \(\approx\) \(1.038194967 + 0.03880145379i\)

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 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + iT \)
53 \( 1 - iT \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
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

−24.476315726269092309884469514394, −22.82680359223919855712172114458, −21.90949548817969685583018026386, −21.11378128381052249962008378226, −20.53700442204814807174240261820, −19.60237271758153739149957771589, −18.97617962047896122419726697348, −18.21597751357005447015870713567, −16.93309284942283331269458035964, −16.28802208162510848175768216205, −15.51911567853692965770885480985, −14.25545158591194816502497597259, −13.62713003052638415056603544773, −12.29792336186900702648146935949, −11.53003203946642371754163722145, −10.295169803216665571350235511538, −9.84670780614620220140189023317, −8.74455991916840809259013415675, −8.15869194907097199394558172586, −7.210490026031644886949949966408, −5.842907299989230929545209758243, −4.22436194972336298364488388147, −3.37452576046263424423116566788, −2.46155795464991855861164448994, −1.16166448841896705453656028311, 1.16631364217087936483790693701, 2.10417625219516973361363239062, 3.365802312833563634660698182393, 4.83778524354198385783454314213, 6.21030491441727425112383639557, 7.063411831487545016232823734032, 7.81493056026633917523002259761, 8.702152196171579679971757061340, 9.57219384117477475064251417608, 10.26135736924382400045417562308, 11.67854190383374195094008662937, 12.512498146690089206042016027932, 13.82709026534217264308805110809, 14.46470630259091840058940135312, 15.33995096922728290653929385577, 16.07487829605375704277393698291, 17.4437764863916124962696141416, 17.79147290530978626003881752580, 18.93256290933661191560788669730, 19.50669103047726594101595855176, 20.26625081458625472369620884647, 21.02554382794855694213001485152, 22.382494800441148399258181245083, 23.56497388929639445802305596729, 24.12601078708732325873415626701

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