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Properties

Label 2-1584-33.8-c1-0-12
Degree $2$
Conductor $1584$
Sign $0.344 + 0.938i$
Analytic cond. $12.6483$
Root an. cond. $3.55644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 + 2.36i)5-s + (−2.58 − 0.840i)7-s + (−2.71 + 1.89i)11-s + (−1.67 − 2.31i)13-s + (3.60 + 2.62i)17-s + (1.81 − 0.590i)19-s − 0.816i·23-s + (−1.09 − 3.36i)25-s + (2.95 − 9.07i)29-s + (−4.84 + 3.51i)31-s + (6.42 − 4.66i)35-s + (1.83 − 5.65i)37-s + (2.60 + 8.02i)41-s − 11.8i·43-s + (7.34 − 2.38i)47-s + ⋯
L(s)  = 1  + (−0.768 + 1.05i)5-s + (−0.977 − 0.317i)7-s + (−0.820 + 0.572i)11-s + (−0.465 − 0.640i)13-s + (0.875 + 0.636i)17-s + (0.416 − 0.135i)19-s − 0.170i·23-s + (−0.218 − 0.672i)25-s + (0.547 − 1.68i)29-s + (−0.869 + 0.631i)31-s + (1.08 − 0.789i)35-s + (0.302 − 0.929i)37-s + (0.406 + 1.25i)41-s − 1.81i·43-s + (1.07 − 0.347i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $0.344 + 0.938i$
Analytic conductor: \(12.6483\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :1/2),\ 0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6658333898\)
\(L(\frac12)\) \(\approx\) \(0.6658333898\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (2.71 - 1.89i)T \)
good5 \( 1 + (1.71 - 2.36i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (2.58 + 0.840i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.67 + 2.31i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.60 - 2.62i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.81 + 0.590i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.816iT - 23T^{2} \)
29 \( 1 + (-2.95 + 9.07i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.84 - 3.51i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.83 + 5.65i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.60 - 8.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.8iT - 43T^{2} \)
47 \( 1 + (-7.34 + 2.38i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.14 + 8.45i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0887 + 0.0288i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.26 + 1.73i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.40T + 67T^{2} \)
71 \( 1 + (-2.12 + 2.91i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.67 + 0.543i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.19 - 5.77i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.33 - 4.60i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.06iT - 89T^{2} \)
97 \( 1 + (1.35 - 0.981i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.523619341623474947927224262274, −8.203710193237182331538966535610, −7.56447729247657754783253991346, −7.03068439248581340091768702955, −6.12012483603587267506569891252, −5.18284812707935138456152045970, −3.93229150489442937776719337331, −3.25817381804961091623017622763, −2.39088012593730503477960505624, −0.31041881964104066094136878933, 1.02345187316975734947982343974, 2.74046556138507689389662662757, 3.55268251275692591184205096505, 4.67852825804230472961974853381, 5.35809409055064319277393234870, 6.26661195105212430510315410063, 7.39773939990720723070054009349, 7.911252186547750708954689324836, 8.953676973031175088628711979726, 9.331779359826551627684441713375

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