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Properties

Label 2-588-28.27-c1-0-36
Degree $2$
Conductor $588$
Sign $-0.195 + 0.980i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (1.33 − 0.458i)2-s − 3-s + (1.57 − 1.22i)4-s − 2.45i·5-s + (−1.33 + 0.458i)6-s + (1.55 − 2.36i)8-s + 9-s + (−1.12 − 3.28i)10-s + 1.26i·11-s + (−1.57 + 1.22i)12-s − 2.99i·13-s + 2.45i·15-s + (0.992 − 3.87i)16-s + 1.83i·17-s + (1.33 − 0.458i)18-s − 4.15·19-s + ⋯
L(s)  = 1  + (0.946 − 0.324i)2-s − 0.577·3-s + (0.789 − 0.613i)4-s − 1.09i·5-s + (−0.546 + 0.187i)6-s + (0.548 − 0.836i)8-s + 0.333·9-s + (−0.355 − 1.03i)10-s + 0.381i·11-s + (−0.456 + 0.354i)12-s − 0.831i·13-s + 0.633i·15-s + (0.248 − 0.968i)16-s + 0.444i·17-s + (0.315 − 0.108i)18-s − 0.954·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.195 + 0.980i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32330 - 1.61356i\)
\(L(\frac12)\) \(\approx\) \(1.32330 - 1.61356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.458i)T \)
3 \( 1 + T \)
7 \( 1 \)
good5 \( 1 + 2.45iT - 5T^{2} \)
11 \( 1 - 1.26iT - 11T^{2} \)
13 \( 1 + 2.99iT - 13T^{2} \)
17 \( 1 - 1.83iT - 17T^{2} \)
19 \( 1 + 4.15T + 19T^{2} \)
23 \( 1 + 6.73iT - 23T^{2} \)
29 \( 1 + 9.42T + 29T^{2} \)
31 \( 1 - 9.43T + 31T^{2} \)
37 \( 1 - 7.51T + 37T^{2} \)
41 \( 1 - 1.08iT - 41T^{2} \)
43 \( 1 - 6.27iT - 43T^{2} \)
47 \( 1 - 7.35T + 47T^{2} \)
53 \( 1 + 0.0716T + 53T^{2} \)
59 \( 1 - 3.36T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 - 2.80iT - 67T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 - 8.10iT - 73T^{2} \)
79 \( 1 + 1.78iT - 79T^{2} \)
83 \( 1 - 5.33T + 83T^{2} \)
89 \( 1 + 8.57iT - 89T^{2} \)
97 \( 1 - 7.10iT - 97T^{2} \)
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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

−10.59376449787913113143151655546, −9.897542268253071365488943558045, −8.708327511361108072139574932260, −7.67620528645340914314802293706, −6.44917928555343315135761530915, −5.70599641473385583693355791527, −4.73052839133518140024870343425, −4.11887612463788506301512077712, −2.47245161383818608393245166162, −0.966258254836521024658311894163, 2.15717433418235889254919772390, 3.40216865182157274744713543198, 4.38070020842749394705289638167, 5.56629858166727757254478810140, 6.37358366549327774479425162972, 7.05156008537418467697288703704, 7.88363712819597602269176031182, 9.256590489019618003992199640096, 10.42812745982299519484089814313, 11.23641715600634973371661268519

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