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Properties

Label 2-588-7.6-c2-0-1
Degree $2$
Conductor $588$
Sign $-0.987 + 0.156i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s + 5.82i·5-s − 2.99·9-s − 6.86·11-s − 3.62i·13-s − 10.0·15-s + 9.71i·17-s + 30.2i·19-s − 18.1·23-s − 8.96·25-s − 5.19i·27-s − 40.4·29-s − 55.2i·31-s − 11.8i·33-s + 54.9·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.16i·5-s − 0.333·9-s − 0.623·11-s − 0.278i·13-s − 0.672·15-s + 0.571i·17-s + 1.59i·19-s − 0.789·23-s − 0.358·25-s − 0.192i·27-s − 1.39·29-s − 1.78i·31-s − 0.360i·33-s + 1.48·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.987 + 0.156i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ -0.987 + 0.156i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7973027594\)
\(L(\frac12)\) \(\approx\) \(0.7973027594\)
\(L(2)\) 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 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 5.82iT - 25T^{2} \)
11 \( 1 + 6.86T + 121T^{2} \)
13 \( 1 + 3.62iT - 169T^{2} \)
17 \( 1 - 9.71iT - 289T^{2} \)
19 \( 1 - 30.2iT - 361T^{2} \)
23 \( 1 + 18.1T + 529T^{2} \)
29 \( 1 + 40.4T + 841T^{2} \)
31 \( 1 + 55.2iT - 961T^{2} \)
37 \( 1 - 54.9T + 1.36e3T^{2} \)
41 \( 1 + 56.3iT - 1.68e3T^{2} \)
43 \( 1 + 66.0T + 1.84e3T^{2} \)
47 \( 1 - 49.4iT - 2.20e3T^{2} \)
53 \( 1 + 81.0T + 2.80e3T^{2} \)
59 \( 1 - 34.7iT - 3.48e3T^{2} \)
61 \( 1 + 0.0382iT - 3.72e3T^{2} \)
67 \( 1 + 64.0T + 4.48e3T^{2} \)
71 \( 1 - 50.2T + 5.04e3T^{2} \)
73 \( 1 - 21.3iT - 5.32e3T^{2} \)
79 \( 1 + 47.5T + 6.24e3T^{2} \)
83 \( 1 + 33.6iT - 6.88e3T^{2} \)
89 \( 1 - 156. iT - 7.92e3T^{2} \)
97 \( 1 + 43.7iT - 9.40e3T^{2} \)
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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

−10.81280382053835974873101836114, −10.15777291697792721944193018255, −9.488976476815159308202552852111, −8.101516149567430092901268021803, −7.58141334770850107281975254163, −6.23197143671717115032526674628, −5.64414285449423761387542689486, −4.16870167281353979457084537574, −3.31898759028256837263219273069, −2.12707967886313206155293603402, 0.28742722035082587005500859593, 1.64994694287480785424020215450, 3.01748028493011765658794885784, 4.62239503741742632275451831596, 5.23464966696944469710061302277, 6.44706401683014297927748442590, 7.41299656541601152734452953329, 8.311302998599150146428428541420, 9.040264570472398448804061948995, 9.851972074804723128733444061129

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