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Properties

Label 2-1584-33.17-c1-0-12
Degree $2$
Conductor $1584$
Sign $0.791 - 0.610i$
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  + (2.13 + 0.694i)5-s + (2.38 + 3.27i)7-s + (3.31 + 0.0200i)11-s + (4.42 − 1.43i)13-s + (−0.0235 + 0.0725i)17-s + (−1.40 + 1.93i)19-s − 3.22i·23-s + (0.0437 + 0.0318i)25-s + (1.48 − 1.08i)29-s + (−0.517 − 1.59i)31-s + (2.81 + 8.66i)35-s + (−5.87 + 4.27i)37-s + (−6.82 − 4.96i)41-s + 4.28i·43-s + (3.65 − 5.02i)47-s + ⋯
L(s)  = 1  + (0.956 + 0.310i)5-s + (0.899 + 1.23i)7-s + (0.999 + 0.00604i)11-s + (1.22 − 0.398i)13-s + (−0.00571 + 0.0175i)17-s + (−0.323 + 0.444i)19-s − 0.672i·23-s + (0.00875 + 0.00636i)25-s + (0.276 − 0.200i)29-s + (−0.0929 − 0.286i)31-s + (0.475 + 1.46i)35-s + (−0.966 + 0.702i)37-s + (−1.06 − 0.774i)41-s + 0.652i·43-s + (0.532 − 0.733i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $0.791 - 0.610i$
Analytic conductor: \(12.6483\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :1/2),\ 0.791 - 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506953430\)
\(L(\frac12)\) \(\approx\) \(2.506953430\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 - 0.0200i)T \)
good5 \( 1 + (-2.13 - 0.694i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.38 - 3.27i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-4.42 + 1.43i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.0235 - 0.0725i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.40 - 1.93i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.22iT - 23T^{2} \)
29 \( 1 + (-1.48 + 1.08i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.517 + 1.59i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.87 - 4.27i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.82 + 4.96i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.28iT - 43T^{2} \)
47 \( 1 + (-3.65 + 5.02i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.16 + 0.379i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.341 - 0.469i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.59 + 1.16i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + (1.06 + 0.346i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.82 - 10.7i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.627 - 0.203i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.15 + 9.71i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.58iT - 89T^{2} \)
97 \( 1 + (-5.08 - 15.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.410074652561073597155512474020, −8.600762230749513001668270824147, −8.286281503422685013033193766492, −6.89915702218525366865907600588, −6.06733928098468228752792514320, −5.65818462924922418145914036138, −4.58703198946206496691507754721, −3.43399524180939237785102765966, −2.24091229606545263414708710677, −1.47608339544318543392058280599, 1.19045727859764959783124206199, 1.78708486964782220719283296267, 3.51454230177834095747853053772, 4.27909371857526441221616670135, 5.17456253709336637890056410185, 6.15424219513892533927108420785, 6.86270675350885450173622378679, 7.71834557217400498550089484202, 8.737008822798641997157703246612, 9.195973782895714005102583154705

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