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Properties

Label 1-609-609.74-r1-0-0
Degree $1$
Conductor $609$
Sign $0.0321 - 0.999i$
Analytic cond. $65.4461$
Root an. cond. $65.4461$
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  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (0.988 − 0.149i)5-s + (0.900 − 0.433i)8-s + (0.955 − 0.294i)10-s + (−0.0747 − 0.997i)11-s + (−0.900 − 0.433i)13-s + (0.826 − 0.563i)16-s + (0.5 + 0.866i)17-s + (−0.733 − 0.680i)19-s + (0.900 − 0.433i)20-s + (−0.222 − 0.974i)22-s + (−0.365 − 0.930i)23-s + (0.955 − 0.294i)25-s + (−0.955 − 0.294i)26-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (0.988 − 0.149i)5-s + (0.900 − 0.433i)8-s + (0.955 − 0.294i)10-s + (−0.0747 − 0.997i)11-s + (−0.900 − 0.433i)13-s + (0.826 − 0.563i)16-s + (0.5 + 0.866i)17-s + (−0.733 − 0.680i)19-s + (0.900 − 0.433i)20-s + (−0.222 − 0.974i)22-s + (−0.365 − 0.930i)23-s + (0.955 − 0.294i)25-s + (−0.955 − 0.294i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 609 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0321 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 609 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0321 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(609\)    =    \(3 \cdot 7 \cdot 29\)
Sign: $0.0321 - 0.999i$
Analytic conductor: \(65.4461\)
Root analytic conductor: \(65.4461\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{609} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 609,\ (1:\ ),\ 0.0321 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.365231314 - 3.258674702i\)
\(L(\frac12)\) \(\approx\) \(3.365231314 - 3.258674702i\)
\(L(1)\) \(\approx\) \(2.155229887 - 0.7544451435i\)
\(L(1)\) \(\approx\) \(2.155229887 - 0.7544451435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.988 - 0.149i)T \)
5 \( 1 + (0.988 - 0.149i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.733 - 0.680i)T \)
23 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (0.365 - 0.930i)T \)
37 \( 1 + (0.0747 - 0.997i)T \)
41 \( 1 - T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (-0.365 + 0.930i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (0.826 + 0.563i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.988 - 0.149i)T \)
79 \( 1 + (0.0747 - 0.997i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−22.97181996687773402251872515803, −22.24009620885946526254369968763, −21.46504542891756427658643553720, −20.85042550328174293574567851432, −20.02715650142650396356189343598, −19.02363320331360693489593317065, −17.85580738526697148282587416836, −17.13532982856963624338003804548, −16.355802877060595604998913713460, −15.28280802641710383886753308674, −14.483899151410610766903020864532, −13.941133947136295789050080139551, −12.95499879985062175883271603514, −12.25304353623730941913116106652, −11.38825933242472343099624502226, −10.10612379063521768407220551290, −9.6922503963434043733900824687, −8.14857100330440614937063700846, −7.072788856426384742457353850493, −6.46837041065472019014096191011, −5.28045617945716809963575885023, −4.760680457731896104527601217967, −3.43837418448896132461231556364, −2.34744897116809461826306134826, −1.60646505766033300888703308693, 0.71569488529632534952654142490, 2.07832569662484224144721644510, 2.81129835114143213676652208335, 4.03381814053834748095558656897, 5.09229073636901532696075368571, 5.88323721804825148820334641614, 6.55838864196230173487560508885, 7.79594508776485522503146973066, 8.90702292638254653193564855694, 10.16014362709359943510825357254, 10.6425841064723388262083586983, 11.77357877407722656876654655207, 12.776324750579643184758585893876, 13.24311385730710276855061945866, 14.27300405739358166081338708789, 14.76926197226442213373526324801, 15.869807537971796490030891102583, 16.82437164638194149180537634493, 17.383712015991235253530917836055, 18.72584574816420889478423081358, 19.46497794220870667444982523295, 20.4364168623569957585059893635, 21.20348261763537357154808592364, 21.87661584122719981596590723008, 22.35594712126255659917257093566

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