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Properties

Label 2-588-196.27-c1-0-10
Degree $2$
Conductor $588$
Sign $-0.118 - 0.992i$
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  + (−1.41 − 0.0505i)2-s + (−0.222 + 0.974i)3-s + (1.99 + 0.142i)4-s + (2.57 + 0.587i)5-s + (0.363 − 1.36i)6-s + (−2.51 − 0.835i)7-s + (−2.81 − 0.302i)8-s + (−0.900 − 0.433i)9-s + (−3.60 − 0.960i)10-s + (1.03 + 2.15i)11-s + (−0.583 + 1.91i)12-s + (1.89 + 3.94i)13-s + (3.50 + 1.30i)14-s + (−1.14 + 2.37i)15-s + (3.95 + 0.570i)16-s + (−0.584 + 0.466i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0357i)2-s + (−0.128 + 0.562i)3-s + (0.997 + 0.0714i)4-s + (1.15 + 0.262i)5-s + (0.148 − 0.557i)6-s + (−0.948 − 0.315i)7-s + (−0.994 − 0.107i)8-s + (−0.300 − 0.144i)9-s + (−1.14 − 0.303i)10-s + (0.312 + 0.649i)11-s + (−0.168 + 0.552i)12-s + (0.526 + 1.09i)13-s + (0.936 + 0.349i)14-s + (−0.295 + 0.614i)15-s + (0.989 + 0.142i)16-s + (−0.141 + 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.118 - 0.992i$
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: \((2,\ 588,\ (\ :1/2),\ -0.118 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.592399 + 0.667525i\)
\(L(\frac12)\) \(\approx\) \(0.592399 + 0.667525i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0505i)T \)
3 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (2.51 + 0.835i)T \)
good5 \( 1 + (-2.57 - 0.587i)T + (4.50 + 2.16i)T^{2} \)
11 \( 1 + (-1.03 - 2.15i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.89 - 3.94i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (0.584 - 0.466i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 + (-6.32 - 5.04i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-1.27 - 1.60i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 - 0.240T + 31T^{2} \)
37 \( 1 + (1.99 + 2.50i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-11.4 - 2.60i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (9.66 - 2.20i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-2.94 + 1.41i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (7.68 - 9.63i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-0.367 - 1.61i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (1.02 - 0.820i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 - 9.89iT - 67T^{2} \)
71 \( 1 + (-2.19 - 1.75i)T + (15.7 + 69.2i)T^{2} \)
73 \( 1 + (4.78 - 9.92i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + 6.28iT - 79T^{2} \)
83 \( 1 + (-5.85 - 2.82i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-6.43 + 13.3i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + 11.5iT - 97T^{2} \)
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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

−10.67478465334245742012240504836, −9.894767103094055613448041693047, −9.342234873344670686703149641137, −8.755646427530667330497010787760, −7.19388981718739428409133996896, −6.51936617705383455001514806869, −5.82312135530775821797482852684, −4.22000654367748133518660190640, −2.89847525789626444994995907104, −1.64102743001027344221679574137, 0.69987882685088301697825805551, 2.17365442248723818173939192594, 3.19095286832546766920641323513, 5.36781779109539661912027852948, 6.32623467421040107281577714514, 6.56614608555851722038181855365, 8.032700748309596910654191547854, 8.810749644530168449949738248536, 9.427848791107237740216341711743, 10.44967799825078975495767924472

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