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Properties

Label 2-588-196.27-c1-0-9
Degree $2$
Conductor $588$
Sign $0.998 - 0.0581i$
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  + (−1.16 − 0.803i)2-s + (−0.222 + 0.974i)3-s + (0.707 + 1.87i)4-s + (−3.84 − 0.877i)5-s + (1.04 − 0.955i)6-s + (−2.22 − 1.42i)7-s + (0.681 − 2.74i)8-s + (−0.900 − 0.433i)9-s + (3.76 + 4.11i)10-s + (−0.211 − 0.438i)11-s + (−1.98 + 0.273i)12-s + (1.80 + 3.74i)13-s + (1.44 + 3.44i)14-s + (1.71 − 3.55i)15-s + (−2.99 + 2.64i)16-s + (3.87 − 3.09i)17-s + ⋯
L(s)  = 1  + (−0.822 − 0.568i)2-s + (−0.128 + 0.562i)3-s + (0.353 + 0.935i)4-s + (−1.72 − 0.392i)5-s + (0.425 − 0.390i)6-s + (−0.842 − 0.538i)7-s + (0.240 − 0.970i)8-s + (−0.300 − 0.144i)9-s + (1.19 + 1.30i)10-s + (−0.0636 − 0.132i)11-s + (−0.571 + 0.0788i)12-s + (0.500 + 1.03i)13-s + (0.387 + 0.921i)14-s + (0.441 − 0.917i)15-s + (−0.749 + 0.661i)16-s + (0.940 − 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.998 - 0.0581i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.998 - 0.0581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515444 + 0.0150037i\)
\(L(\frac12)\) \(\approx\) \(0.515444 + 0.0150037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.803i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (2.22 + 1.42i)T \)
good5 \( 1 + (3.84 + 0.877i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (0.211 + 0.438i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.80 - 3.74i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (-3.87 + 3.09i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 + (1.01 + 0.811i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-3.84 - 4.82i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 + (-5.94 - 7.45i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (0.785 + 0.179i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-9.95 + 2.27i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (7.29 - 3.51i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-4.02 + 5.04i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-0.246 - 1.08i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-4.52 + 3.61i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 - 5.76iT - 67T^{2} \)
71 \( 1 + (-3.14 - 2.50i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (-6.35 + 13.1i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + 17.2iT - 79T^{2} \)
83 \( 1 + (-4.95 - 2.38i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (6.19 - 12.8i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + 3.59iT - 97T^{2} \)
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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

−10.76825281128217491010744977459, −9.817783008876043014661909713870, −9.022969452647833698503584299110, −8.217621313760752346759137986883, −7.38379412765693043210611083516, −6.52054108609485274936907860969, −4.64757461432863378656715045790, −3.83385182867525040135041120095, −3.11262028758199367096506515437, −0.78866932031408659354502187801, 0.62682383011510327771366084587, 2.71682592454294173700005389152, 3.92357493370515392429334786862, 5.57851073852899993927734600545, 6.35849177809396531940492648108, 7.34263590388408425261923918335, 7.992769848331820490988056060382, 8.539259250777412139536087897179, 9.782357888057360006291462259315, 10.69386039853356124996269986654

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