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Properties

Label 1-675-675.23-r0-0-0
Degree $1$
Conductor $675$
Sign $0.00930 - 0.999i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (0.529 + 0.848i)2-s + (−0.438 + 0.898i)4-s + (−0.984 − 0.173i)7-s + (−0.994 + 0.104i)8-s + (−0.0348 − 0.999i)11-s + (−0.529 + 0.848i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.829 − 0.559i)22-s + (0.139 − 0.990i)23-s − 26-s + (0.587 − 0.809i)28-s + (−0.241 − 0.970i)29-s + ⋯
L(s)  = 1  + (0.529 + 0.848i)2-s + (−0.438 + 0.898i)4-s + (−0.984 − 0.173i)7-s + (−0.994 + 0.104i)8-s + (−0.0348 − 0.999i)11-s + (−0.529 + 0.848i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.829 − 0.559i)22-s + (0.139 − 0.990i)23-s − 26-s + (0.587 − 0.809i)28-s + (−0.241 − 0.970i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00930 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00930 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.00930 - 0.999i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.00930 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1282428649 - 0.1270546478i\)
\(L(\frac12)\) \(\approx\) \(0.1282428649 - 0.1270546478i\)
\(L(1)\) \(\approx\) \(0.7639118757 + 0.3577900787i\)
\(L(1)\) \(\approx\) \(0.7639118757 + 0.3577900787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.529 + 0.848i)T \)
7 \( 1 + (-0.984 - 0.173i)T \)
11 \( 1 + (-0.0348 - 0.999i)T \)
13 \( 1 + (-0.529 + 0.848i)T \)
17 \( 1 + (-0.406 + 0.913i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (0.139 - 0.990i)T \)
29 \( 1 + (-0.241 - 0.970i)T \)
31 \( 1 + (-0.719 - 0.694i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (-0.848 - 0.529i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.694 - 0.719i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (-0.882 - 0.469i)T \)
67 \( 1 + (0.970 + 0.241i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (-0.743 + 0.669i)T \)
79 \( 1 + (0.241 + 0.970i)T \)
83 \( 1 + (-0.0697 + 0.997i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (0.275 - 0.961i)T \)
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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

−22.74235045015576027597638505814, −22.1961791988889169392937787219, −21.46603570782534809398694960260, −20.16051195001202459708565213678, −20.04681996510013441574707174632, −19.1220021330213394481690307579, −18.09250436021077649983838868492, −17.54789744877373118478234315394, −16.12494827006930737101009492149, −15.361926640019587946760445794357, −14.67090657124199754268914422463, −13.473651260816980243704904790369, −12.952930317184261688257366959352, −12.188555842110710557352696920519, −11.330512536092546058219135768283, −10.28223287591958839252349156353, −9.61621434952439398926204546893, −8.927717833829796014241440302021, −7.34587758393836083553910886810, −6.55551453841262875441658126685, −5.28891936562399101986530640491, −4.73150751217133362594985706116, −3.32740400975208251125932181457, −2.79192722375557864904222945263, −1.543112721931446707372746981677, 0.069380423234969512169798230553, 2.21169230323186325970743718374, 3.50840911398879826674975774246, 4.0575185163800822600828036838, 5.363035699901945329455984928396, 6.23860449184586505378287987515, 6.82743888261567445532060500523, 7.94629472383583606733294190692, 8.78718899778069848368506881163, 9.66332172561171784846692857502, 10.78318206617136458273028983072, 11.97724352860337499397966672669, 12.687229187205579565977679528639, 13.546213479600863397036108986057, 14.215259562823025714033333256585, 15.13353361203868027259839696755, 16.00464206559009499893786539098, 16.76061096036969763952195826713, 17.13037046019835190280991915737, 18.626152350934431098742864020423, 19.00052940697207612066005996086, 20.19465387751861112107043019337, 21.21594220934585685286740737566, 21.95369404168799654951420409181, 22.55790008203230475629940897768

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