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Properties

Label 1-3528-3528.5-r0-0-0
Degree $1$
Conductor $3528$
Sign $-0.974 - 0.225i$
Analytic cond. $16.3839$
Root an. cond. $16.3839$
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  + (−0.365 − 0.930i)5-s + (0.955 − 0.294i)11-s + (0.955 − 0.294i)13-s + (0.0747 − 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.826 + 0.563i)23-s + (−0.733 + 0.680i)25-s + (0.0747 − 0.997i)29-s − 31-s + (−0.826 − 0.563i)37-s + (−0.988 − 0.149i)41-s + (0.988 − 0.149i)43-s + (−0.222 + 0.974i)47-s + (0.826 − 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)5-s + (0.955 − 0.294i)11-s + (0.955 − 0.294i)13-s + (0.0747 − 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.826 + 0.563i)23-s + (−0.733 + 0.680i)25-s + (0.0747 − 0.997i)29-s − 31-s + (−0.826 − 0.563i)37-s + (−0.988 − 0.149i)41-s + (0.988 − 0.149i)43-s + (−0.222 + 0.974i)47-s + (0.826 − 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.974 - 0.225i$
Analytic conductor: \(16.3839\)
Root analytic conductor: \(16.3839\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3528,\ (0:\ ),\ -0.974 - 0.225i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1096331642 - 0.9576884165i\)
\(L(\frac12)\) \(\approx\) \(0.1096331642 - 0.9576884165i\)
\(L(1)\) \(\approx\) \(0.8766055145 - 0.3418557755i\)
\(L(1)\) \(\approx\) \(0.8766055145 - 0.3418557755i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.365 - 0.930i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.826 + 0.563i)T \)
29 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.955 - 0.294i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−18.93488842201430528341280686048, −18.43519324326408849817632592529, −17.841615933442013780189140998925, −16.85302578401385614020776793292, −16.438369075340406092218456201850, −15.44745455970152406029218969152, −14.91262198707383475494988007568, −14.275683700330359894392681477946, −13.74551152580888502375897782008, −12.66353708031103812877992265282, −12.088431576564515254075297629857, −11.39748691683629928484010788488, −10.503046679357419550298085358873, −10.30080011282446472695065131785, −9.08911755714181451828031912299, −8.50722138807124881819010456192, −7.7199997033335578195032132094, −6.839971788961553471513849097322, −6.3333192924383914645036863253, −5.674426328900379903948517323658, −4.35037957457492514004980216293, −3.80918692762002672851744147385, −3.23134529631218389761182864059, −1.98668994497695434355275973856, −1.43283300352166992237897266271, 0.28718233495788146320039007864, 1.19942195472134668339807211843, 2.0503012411169910228109173199, 3.26936994613448900951285240054, 3.94216486259679295493579798778, 4.623551656947223840342044248404, 5.53092290676321687728423793197, 6.153043160439909942025216045836, 7.11536221231245913249350456798, 7.83441100091976509080812410500, 8.71720122132875768138121258129, 9.079692840806542069761847527395, 9.84660818567222540315089377887, 10.93107673422131130148451012096, 11.50103796601640555487189127800, 12.10561877969249059871169517471, 12.86871275018481669541553189233, 13.627841337209863569127958008353, 14.06515573044854126152627193419, 15.16277307392823809868211870095, 15.761514191963634361465228607644, 16.28675072708897315825162521408, 17.026369238289142393304972826, 17.64610046329141915591987460752, 18.37427028877570984749492360149

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