Properties

Label 2-1584-33.32-c1-0-5
Degree $2$
Conductor $1584$
Sign $0.394 - 0.918i$
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  + 1.41i·5-s − 2.44i·7-s + (−1.73 − 2.82i)11-s + 4.89i·13-s + 7.34i·19-s + 2.82i·23-s + 2.99·25-s − 6.92·29-s + 4·31-s + 3.46·35-s + 8·37-s + 6.92·41-s − 2.44i·43-s + 2.82i·47-s + 1.00·49-s + ⋯
L(s)  = 1  + 0.632i·5-s − 0.925i·7-s + (−0.522 − 0.852i)11-s + 1.35i·13-s + 1.68i·19-s + 0.589i·23-s + 0.599·25-s − 1.28·29-s + 0.718·31-s + 0.585·35-s + 1.31·37-s + 1.08·41-s − 0.373i·43-s + 0.412i·47-s + 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1584\)    =    \(2^{4} \cdot 3^{2} \cdot 11\)
Sign: $0.394 - 0.918i$
Analytic conductor: \(12.6483\)
Root analytic conductor: \(3.55644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1584} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1584,\ (\ :1/2),\ 0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.425582212\)
\(L(\frac12)\) \(\approx\) \(1.425582212\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 2.82i)T \)
good5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 + 2.44iT - 7T^{2} \)
13 \( 1 - 4.89iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 7.34iT - 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 2.44iT - 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 - 9.89iT - 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 + 4.89iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 12.2iT - 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.660089318908678122407795664307, −8.789177366703760277342991092267, −7.77277563016265024545729219967, −7.29416779965492839831096218492, −6.31309543869948257126077744646, −5.66169560283967870416256574274, −4.31840155723861673376304135168, −3.70626333580397874761226586116, −2.59800640632133392020786523019, −1.25085631993390629872565787967, 0.60537735200468892605965109108, 2.25154868840680760201573838091, 2.99148181967373515006791368642, 4.48001706018860926515985375561, 5.12041973297983471580163025067, 5.83428503259135170463720915718, 6.90804906601667054497472146717, 7.80357205865799048911643788869, 8.503866540749116638175445529034, 9.250329297038510320763228194495

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