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Properties

Label 1-675-675.11-r1-0-0
Degree $1$
Conductor $675$
Sign $0.277 + 0.960i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.939 − 0.342i)7-s + (−0.913 − 0.406i)8-s + (0.374 + 0.927i)11-s + (0.990 − 0.139i)13-s + (0.882 + 0.469i)14-s + (0.848 + 0.529i)16-s + (0.104 + 0.994i)17-s + (0.913 + 0.406i)19-s + (−0.241 − 0.970i)22-s + (−0.0348 − 0.999i)23-s − 26-s + (−0.809 − 0.587i)28-s + (−0.438 + 0.898i)29-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.939 − 0.342i)7-s + (−0.913 − 0.406i)8-s + (0.374 + 0.927i)11-s + (0.990 − 0.139i)13-s + (0.882 + 0.469i)14-s + (0.848 + 0.529i)16-s + (0.104 + 0.994i)17-s + (0.913 + 0.406i)19-s + (−0.241 − 0.970i)22-s + (−0.0348 − 0.999i)23-s − 26-s + (−0.809 − 0.587i)28-s + (−0.438 + 0.898i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(72.5388\)
Root analytic conductor: \(72.5388\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (1:\ ),\ 0.277 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7645285810 + 0.5747201240i\)
\(L(\frac12)\) \(\approx\) \(0.7645285810 + 0.5747201240i\)
\(L(1)\) \(\approx\) \(0.6800842548 + 0.04855176006i\)
\(L(1)\) \(\approx\) \(0.6800842548 + 0.04855176006i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.990 - 0.139i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.374 + 0.927i)T \)
13 \( 1 + (0.990 - 0.139i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.0348 - 0.999i)T \)
29 \( 1 + (-0.438 + 0.898i)T \)
31 \( 1 + (0.559 - 0.829i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (-0.990 + 0.139i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.559 - 0.829i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.374 - 0.927i)T \)
61 \( 1 + (-0.615 + 0.788i)T \)
67 \( 1 + (0.438 + 0.898i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (0.438 - 0.898i)T \)
83 \( 1 + (0.719 + 0.694i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (-0.997 + 0.0697i)T \)
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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

−22.40804499703871091975016677523, −21.37944486919841027504584570611, −20.639923038039634838999482295577, −19.63438594174837117398989029808, −19.094068527709559189178164854388, −18.35025182396641979104429005065, −17.55671218249089222915018055500, −16.4833633172275007401529339000, −15.96958072057469445240658524166, −15.38482755152045811482703094146, −14.00885696609750335027442600830, −13.34979152127353127990531543791, −11.935886511838307136899946934100, −11.473962959421278455752130258168, −10.43053030991815035047924453208, −9.4298919058466965858521217755, −8.98430959441173231146604901271, −7.957138704620987157917729241675, −6.92545066471839098032825370720, −6.1710778469147551749763084860, −5.340086667864419034345117018631, −3.51422298681624224845523525586, −2.874832411821926068822610091212, −1.41459561654032438204686687733, −0.39529352394104931686818628720, 0.92475621702304583083311214504, 1.94848833993984228835033578182, 3.24413742697293829577222494761, 4.01116182325908843269150575752, 5.71237632377909836362502897023, 6.598260638284517190692805887906, 7.299358520566729204457836246, 8.38365860024277435001371987739, 9.168464127497010086811394655328, 10.13530857266741440208879904441, 10.56657526835661411286228824511, 11.794916133499768418309067388515, 12.52434837605678587365128271236, 13.37990157544891772262596003492, 14.65652754428347221043517624139, 15.52716762154121754995038979883, 16.312671289768721927202823098400, 16.97737666167631381771899989637, 17.84603928157310913261379090133, 18.676897152747713314630955363561, 19.335682061514901206876960482355, 20.3900806351293846125430819809, 20.5456423371245145687199375958, 21.89292286352673630675571007427, 22.6727860039489463979362428190

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