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Properties

Label 2-1584-33.2-c1-0-8
Degree $2$
Conductor $1584$
Sign $0.620 - 0.784i$
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  + (−0.0381 + 0.0123i)5-s + (−0.145 + 0.199i)7-s + (3.12 + 1.12i)11-s + (−2.18 − 0.711i)13-s + (1.32 + 4.06i)17-s + (3.64 + 5.01i)19-s − 6.79i·23-s + (−4.04 + 2.93i)25-s + (−4.52 − 3.28i)29-s + (−1.48 + 4.56i)31-s + (0.00305 − 0.00941i)35-s + (3.26 + 2.36i)37-s + (7.76 − 5.64i)41-s + 1.03i·43-s + (6.53 + 9.00i)47-s + ⋯
L(s)  = 1  + (−0.0170 + 0.00554i)5-s + (−0.0548 + 0.0754i)7-s + (0.940 + 0.338i)11-s + (−0.607 − 0.197i)13-s + (0.320 + 0.986i)17-s + (0.836 + 1.15i)19-s − 1.41i·23-s + (−0.808 + 0.587i)25-s + (−0.840 − 0.610i)29-s + (−0.266 + 0.819i)31-s + (0.000517 − 0.00159i)35-s + (0.536 + 0.389i)37-s + (1.21 − 0.881i)41-s + 0.157i·43-s + (0.953 + 1.31i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $0.620 - 0.784i$
Analytic conductor: \(12.6483\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :1/2),\ 0.620 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.628244664\)
\(L(\frac12)\) \(\approx\) \(1.628244664\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 - 1.12i)T \)
good5 \( 1 + (0.0381 - 0.0123i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.145 - 0.199i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.18 + 0.711i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.32 - 4.06i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.64 - 5.01i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.79iT - 23T^{2} \)
29 \( 1 + (4.52 + 3.28i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.48 - 4.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.26 - 2.36i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.76 + 5.64i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.03iT - 43T^{2} \)
47 \( 1 + (-6.53 - 9.00i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-8.52 - 2.77i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.63 - 2.25i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-8.06 + 2.62i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 7.94T + 67T^{2} \)
71 \( 1 + (-3.16 + 1.02i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.96 - 9.58i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.86 - 0.930i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.63 + 5.03i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.54iT - 89T^{2} \)
97 \( 1 + (0.935 - 2.88i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.541205251871850365635933376139, −8.821547401626314060819173444146, −7.83183695917812930459977277191, −7.27175250372451309714223107164, −6.15705530815234141938294971632, −5.61729833827070562674565451968, −4.35809188145121391822354873456, −3.70241364038729557485878221927, −2.43865924193206490985118585089, −1.24685239910342065684860701309, 0.71636799608341365413428586644, 2.16915358290354404233893252569, 3.31464629635388035039626531606, 4.19788013425427391325354263926, 5.25590292792939571025625426037, 5.95530298860565281803094974552, 7.24169126581807297281969500286, 7.35227278091721853352783418183, 8.653178153441707975971773188125, 9.478166342562867789235845044316

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