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Properties

Label 2-588-84.11-c1-0-15
Degree $2$
Conductor $588$
Sign $0.752 - 0.658i$
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  + (−1.34 − 0.443i)2-s + (−1.66 − 0.489i)3-s + (1.60 + 1.19i)4-s + (1.08 − 0.626i)5-s + (2.01 + 1.39i)6-s + (−1.63 − 2.31i)8-s + (2.52 + 1.62i)9-s + (−1.73 + 0.360i)10-s + (−2.52 + 4.38i)11-s + (−2.08 − 2.76i)12-s + 4.41·13-s + (−2.11 + 0.509i)15-s + (1.16 + 3.82i)16-s + (−5.06 − 2.92i)17-s + (−2.66 − 3.30i)18-s + (−1.32 + 0.765i)19-s + ⋯
L(s)  = 1  + (−0.949 − 0.313i)2-s + (−0.959 − 0.282i)3-s + (0.803 + 0.595i)4-s + (0.485 − 0.280i)5-s + (0.822 + 0.569i)6-s + (−0.576 − 0.817i)8-s + (0.840 + 0.542i)9-s + (−0.548 + 0.114i)10-s + (−0.762 + 1.32i)11-s + (−0.602 − 0.798i)12-s + 1.22·13-s + (−0.544 + 0.131i)15-s + (0.291 + 0.956i)16-s + (−1.22 − 0.709i)17-s + (−0.627 − 0.778i)18-s + (−0.304 + 0.175i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.752 - 0.658i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577831 + 0.217271i\)
\(L(\frac12)\) \(\approx\) \(0.577831 + 0.217271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.443i)T \)
3 \( 1 + (1.66 + 0.489i)T \)
7 \( 1 \)
good5 \( 1 + (-1.08 + 0.626i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.52 - 4.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 + (5.06 + 2.92i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.32 - 0.765i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.156 + 0.271i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.53iT - 29T^{2} \)
31 \( 1 + (-4.24 - 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.66 - 4.62i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.97iT - 41T^{2} \)
43 \( 1 + 3.18iT - 43T^{2} \)
47 \( 1 + (-5.74 - 9.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.0 - 6.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.98 + 6.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.151 - 0.262i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.13 - 5.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.53T + 71T^{2} \)
73 \( 1 + (0.707 - 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6 - 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.78T + 83T^{2} \)
89 \( 1 + (4.15 - 2.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−10.76897480165550325475032716799, −10.02563866549123099806556877484, −9.200179836936141239940244521483, −8.202033646622765672710275827918, −7.16180684430210901781179502602, −6.52330656283968625315340475515, −5.41074119635240849362749981882, −4.26597777493307169461605636606, −2.42731642568118573213945742527, −1.28752800538067048306746358415, 0.58108509960016598243020219924, 2.29683693489468978217750463309, 3.99236208990847422644275291106, 5.57695782499185241554732666414, 6.08649022409611955971122361361, 6.77851356324497568732650911048, 8.147256853379764230917070393277, 8.782822861954873717959269274422, 9.888677174605853205038313952390, 10.65891217832609552826610723987

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