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Properties

Label 2-588-28.19-c1-0-36
Degree $2$
Conductor $588$
Sign $-0.894 - 0.446i$
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 − 0.0764i)2-s + (−0.5 − 0.866i)3-s + (1.98 + 0.215i)4-s + (−2.68 − 1.55i)5-s + (0.639 + 1.26i)6-s + (−2.79 − 0.457i)8-s + (−0.499 + 0.866i)9-s + (3.67 + 2.39i)10-s + (4.62 − 2.67i)11-s + (−0.807 − 1.82i)12-s − 3.92i·13-s + 3.10i·15-s + (3.90 + 0.858i)16-s + (−4.92 + 2.84i)17-s + (0.772 − 1.18i)18-s + (−0.0854 + 0.147i)19-s + ⋯
L(s)  = 1  + (−0.998 − 0.0540i)2-s + (−0.288 − 0.499i)3-s + (0.994 + 0.107i)4-s + (−1.20 − 0.694i)5-s + (0.261 + 0.514i)6-s + (−0.986 − 0.161i)8-s + (−0.166 + 0.288i)9-s + (1.16 + 0.758i)10-s + (1.39 − 0.805i)11-s + (−0.232 − 0.528i)12-s − 1.08i·13-s + 0.801i·15-s + (0.976 + 0.214i)16-s + (−1.19 + 0.689i)17-s + (0.182 − 0.279i)18-s + (−0.0195 + 0.0339i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.894 - 0.446i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.894 - 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0470537 + 0.199456i\)
\(L(\frac12)\) \(\approx\) \(0.0470537 + 0.199456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0764i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (2.68 + 1.55i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.62 + 2.67i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.92iT - 13T^{2} \)
17 \( 1 + (4.92 - 2.84i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0854 - 0.147i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.31 + 3.06i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.96T + 29T^{2} \)
31 \( 1 + (-0.765 - 1.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.46 - 7.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 - 2.38iT - 43T^{2} \)
47 \( 1 + (1.14 - 1.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.599 - 1.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.49 + 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.70 + 3.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.35 - 4.82i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.04iT - 71T^{2} \)
73 \( 1 + (8.89 - 5.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (12.4 + 7.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (-5.03 - 2.90i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.32iT - 97T^{2} \)
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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

−10.26940881796443445094673978407, −9.008841987499159482248697985596, −8.364091958642309122450110593773, −7.85964804780820957178102234111, −6.67841224448257026540155048014, −5.99413319213251490643843602609, −4.39507844979518991460120351884, −3.24175684584161967526682972860, −1.45435508334543699114508818118, −0.16931473746611855264526587505, 2.00773759920680742140041267752, 3.63497598116626967178894024473, 4.39661317880158716766515356719, 6.11888404176523327431880305924, 7.03361035456856219998651098545, 7.44938887336842001847574961697, 8.879027368309903969279400696367, 9.281856305641883113586578329927, 10.33853800591562877920472586332, 11.18639211934169757253935407165

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