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Properties

Label 1-669-669.38-r1-0-0
Degree $1$
Conductor $669$
Sign $-0.503 - 0.863i$
Analytic cond. $71.8940$
Root an. cond. $71.8940$
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  + (0.828 − 0.559i)2-s + (0.372 − 0.927i)4-s + (−0.741 − 0.671i)5-s + (0.985 − 0.169i)7-s + (−0.210 − 0.977i)8-s + (−0.990 − 0.141i)10-s + (0.548 + 0.836i)11-s + (0.873 + 0.487i)13-s + (0.721 − 0.691i)14-s + (−0.721 − 0.691i)16-s + (0.450 − 0.892i)17-s + (−0.182 − 0.983i)19-s + (−0.899 + 0.437i)20-s + (0.922 + 0.385i)22-s + (0.951 − 0.306i)23-s + ⋯
L(s)  = 1  + (0.828 − 0.559i)2-s + (0.372 − 0.927i)4-s + (−0.741 − 0.671i)5-s + (0.985 − 0.169i)7-s + (−0.210 − 0.977i)8-s + (−0.990 − 0.141i)10-s + (0.548 + 0.836i)11-s + (0.873 + 0.487i)13-s + (0.721 − 0.691i)14-s + (−0.721 − 0.691i)16-s + (0.450 − 0.892i)17-s + (−0.182 − 0.983i)19-s + (−0.899 + 0.437i)20-s + (0.922 + 0.385i)22-s + (0.951 − 0.306i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.503 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(669\)    =    \(3 \cdot 223\)
Sign: $-0.503 - 0.863i$
Analytic conductor: \(71.8940\)
Root analytic conductor: \(71.8940\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{669} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 669,\ (1:\ ),\ -0.503 - 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.923141445 - 3.346949901i\)
\(L(\frac12)\) \(\approx\) \(1.923141445 - 3.346949901i\)
\(L(1)\) \(\approx\) \(1.561436388 - 1.046836677i\)
\(L(1)\) \(\approx\) \(1.561436388 - 1.046836677i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 \)
good2 \( 1 + (0.828 - 0.559i)T \)
5 \( 1 + (-0.741 - 0.671i)T \)
7 \( 1 + (0.985 - 0.169i)T \)
11 \( 1 + (0.548 + 0.836i)T \)
13 \( 1 + (0.873 + 0.487i)T \)
17 \( 1 + (0.450 - 0.892i)T \)
19 \( 1 + (-0.182 - 0.983i)T \)
23 \( 1 + (0.951 - 0.306i)T \)
29 \( 1 + (-0.960 + 0.279i)T \)
31 \( 1 + (0.993 + 0.112i)T \)
37 \( 1 + (0.998 + 0.0565i)T \)
41 \( 1 + (0.967 - 0.251i)T \)
43 \( 1 + (0.812 + 0.583i)T \)
47 \( 1 + (0.182 - 0.983i)T \)
53 \( 1 + (-0.571 + 0.820i)T \)
59 \( 1 + (0.127 - 0.991i)T \)
61 \( 1 + (-0.0141 - 0.999i)T \)
67 \( 1 + (-0.990 + 0.141i)T \)
71 \( 1 + (-0.922 + 0.385i)T \)
73 \( 1 + (0.155 + 0.987i)T \)
79 \( 1 + (0.265 - 0.964i)T \)
83 \( 1 + (-0.812 + 0.583i)T \)
89 \( 1 + (-0.741 + 0.671i)T \)
97 \( 1 + (-0.951 - 0.306i)T \)
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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

−22.85566348512447820103748979484, −22.27518814295997621203935836955, −21.14677188585245674527521227240, −20.869569596705631625299831520594, −19.548344731542070292307834574537, −18.75561690524350036243080689025, −17.807499219807527051842409436712, −16.91854432246248531769165589526, −16.10062220495245904264925163466, −15.13267725977685652279069549605, −14.69587785433580934880651843782, −13.92658093677150359352664745519, −12.92496627285374925150082310420, −11.92801472008609506659328282836, −11.2373205584663868834357518010, −10.63970219232699686805266800718, −8.843666376632517093859726528809, −8.041906738953757646043469223495, −7.52797879830697617054378098014, −6.19487903616043876873930641619, −5.73100542833084504389908827140, −4.33255054745232520457251604611, −3.698381852922235705939245036292, −2.7533484394091228364762842197, −1.25284798896323746204674272885, 0.76890459989190751847614816498, 1.540147308311125238633294818382, 2.82745471079940753444174301014, 4.14174504188066409714514453287, 4.55215886700940534069162289493, 5.43476556429034422370290793080, 6.772473512410812870600296352817, 7.58221034152099342681701363863, 8.83385143678016794351646301164, 9.56132224263018065708574468636, 11.025700013966431737457431359428, 11.359657565981358562064365747392, 12.21628909198705587849356414780, 13.02746641148226121026813826433, 13.9372158033864178135178103324, 14.770134613795763678796790891979, 15.46847019628390343939522764840, 16.354138024730546893270921303250, 17.3435445967675598663061274238, 18.42176145204701742782127877471, 19.24979455404900570953883096301, 20.17051012179457739424679022792, 20.67126017372777464837539561394, 21.2696700184591483873872236665, 22.3757879939593176935359856858

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