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Properties

Label 1-675-675.286-r0-0-0
Degree $1$
Conductor $675$
Sign $0.960 - 0.277i$
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  + (0.990 + 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

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

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.960 - 0.277i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.960 - 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.573778769 - 0.3647753960i\)
\(L(\frac12)\) \(\approx\) \(2.573778769 - 0.3647753960i\)
\(L(1)\) \(\approx\) \(1.866746029 - 0.05112292493i\)
\(L(1)\) \(\approx\) \(1.866746029 - 0.05112292493i\)

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

−22.85471886404026083278806409400, −22.04793009740202855475835172652, −21.2857864808702952077873950169, −20.409705919907595416276843377882, −19.759972619990074166632843951465, −18.87667599657555238482914118500, −17.959155119277974262214686385106, −16.78136615798126998553366323547, −15.77098231528447415611208918832, −15.54881351651726187801425842403, −14.37360513414004696534000615385, −13.6125764374324199703089274032, −12.52040277360319317881135252859, −12.44787207090501637496819459306, −11.04572411399029577863952442899, −10.37217113428766484032608146866, −9.3820692308078762443582709264, −8.21685260940622557415156699293, −6.963774703948950041811739771327, −6.38037925009001246284287669694, −5.4031665116781720994966152029, −4.423850836029555408089704614628, −3.42062851057013685218849230392, −2.59124967315484736282891284456, −1.385058504445708812927533610808, 1.030742316415937347704263884269, 2.67487396972166589520558645550, 3.377149925965228116218599645791, 4.229791462086183103116574938775, 5.59111374411836355871045576888, 6.04009752080834097852455895378, 7.15246187875732465575696712842, 7.92290381042174008138219118872, 9.18005515375257404114459413083, 10.24562479096618970939528924034, 11.16084009690157007244002895417, 11.89223358184498448911438557381, 12.93991248247827511341870972037, 13.737995663810198896106639423870, 13.96903700578429901030774027702, 15.60515578291392843135319112660, 15.8326690342642071237677841440, 16.60844665036576212161027817099, 17.67513088307102022915692377304, 18.87017386192241933731512383892, 19.51213829013087223337192872165, 20.638784514670393110187685262693, 20.981428168447302428802058916746, 22.13950837426425088900538722660, 22.73696126590207142524168644710

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