↑

Properties

Label 1-675-675.14-r1-0-0
Degree $1$
Conductor $675$
Sign $0.508 - 0.861i$
Analytic cond. $72.5388$
Root an. cond. $72.5388$
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  + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (−0.766 − 0.642i)7-s + (−0.104 − 0.994i)8-s + (0.882 + 0.469i)11-s + (−0.0348 + 0.999i)13-s + (0.615 − 0.788i)14-s + (0.990 − 0.139i)16-s + (0.913 + 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.438 + 0.898i)22-s + (−0.374 − 0.927i)23-s − 26-s + (0.809 + 0.587i)28-s + (−0.961 − 0.275i)29-s + ⋯
L(s)  = 1  + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (−0.766 − 0.642i)7-s + (−0.104 − 0.994i)8-s + (0.882 + 0.469i)11-s + (−0.0348 + 0.999i)13-s + (0.615 − 0.788i)14-s + (0.990 − 0.139i)16-s + (0.913 + 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.438 + 0.898i)22-s + (−0.374 − 0.927i)23-s − 26-s + (0.809 + 0.587i)28-s + (−0.961 − 0.275i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.508 - 0.861i$
Analytic conductor: \(72.5388\)
Root analytic conductor: \(72.5388\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (1:\ ),\ 0.508 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5529275371 - 0.3158107777i\)
\(L(\frac12)\) \(\approx\) \(0.5529275371 - 0.3158107777i\)
\(L(1)\) \(\approx\) \(0.7618642102 + 0.3178174642i\)
\(L(1)\) \(\approx\) \(0.7618642102 + 0.3178174642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.0348 + 0.999i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (0.882 + 0.469i)T \)
13 \( 1 + (-0.0348 + 0.999i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.374 - 0.927i)T \)
29 \( 1 + (-0.961 - 0.275i)T \)
31 \( 1 + (-0.241 + 0.970i)T \)
37 \( 1 + (-0.669 + 0.743i)T \)
41 \( 1 + (-0.0348 + 0.999i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.241 - 0.970i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.882 - 0.469i)T \)
61 \( 1 + (0.848 - 0.529i)T \)
67 \( 1 + (-0.961 + 0.275i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (0.961 + 0.275i)T \)
83 \( 1 + (0.559 - 0.829i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (0.719 - 0.694i)T \)
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

−22.39432593845015236306766263671, −22.038813745310919983394398178841, −20.92695688946202229632746593132, −20.325388534703674712713401400044, −19.238075700226051974084303070618, −18.96008854841754870335196051109, −17.977492888265402110266599371258, −17.069509729540614116284062744, −16.1843326264280251393555038054, −15.06485479232388554038840612162, −14.24088775380141821549413391873, −13.35072466642645719417713973237, −12.45044140696003124616114741164, −11.94873708152075560753074810290, −10.95795028936901765933074369711, −9.961789167337406080347236266450, −9.367443728813712905399222027016, −8.46658910084059144529951936655, −7.42604969657036682192496706170, −5.82613633950624924792736017034, −5.49969914510357548328122332479, −3.82288277506381931010707743188, −3.37241411686877493515858045914, −2.21509787954906668984674714350, −1.03662152353566143398422910471, 0.17290945955204398109069520508, 1.511310624238648096916685696633, 3.30233747750488590830920993451, 4.15010127357081093615861353865, 4.99729690987128425495539042324, 6.41313044492782868760134049355, 6.70435979186343640106489698534, 7.68809865748861157655773560503, 8.79778401056185595805251136468, 9.55234079848522212394668129674, 10.28080078635355298137548977859, 11.65121507188141487891579710796, 12.63889576190581821657429619944, 13.39463881579289291684175599640, 14.32186954590805665365204190972, 14.84405172087332593136276062381, 16.00756811108405192397984145508, 16.645134051460119191813340320619, 17.18172636031735539969952806944, 18.17116573823837074468615278856, 19.15328721946608478037207622792, 19.69178808060555451715587312123, 20.88110291851679422306527887318, 21.968945622222474497907163436136, 22.483414757951339128552945554368

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