Properties

Label 2-588-28.19-c1-0-23
Degree $2$
Conductor $588$
Sign $-0.107 + 0.994i$
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  + (0.121 − 1.40i)2-s + (−0.5 − 0.866i)3-s + (−1.97 − 0.341i)4-s + (2.88 + 1.66i)5-s + (−1.28 + 0.599i)6-s + (−0.719 + 2.73i)8-s + (−0.499 + 0.866i)9-s + (2.69 − 3.86i)10-s + (0.810 − 0.468i)11-s + (0.689 + 1.87i)12-s − 1.87i·13-s − 3.33i·15-s + (3.76 + 1.34i)16-s + (4.51 − 2.60i)17-s + (1.15 + 0.809i)18-s + (3.56 − 6.16i)19-s + ⋯
L(s)  = 1  + (0.0855 − 0.996i)2-s + (−0.288 − 0.499i)3-s + (−0.985 − 0.170i)4-s + (1.29 + 0.745i)5-s + (−0.522 + 0.244i)6-s + (−0.254 + 0.967i)8-s + (−0.166 + 0.288i)9-s + (0.853 − 1.22i)10-s + (0.244 − 0.141i)11-s + (0.199 + 0.541i)12-s − 0.519i·13-s − 0.861i·15-s + (0.941 + 0.336i)16-s + (1.09 − 0.631i)17-s + (0.273 + 0.190i)18-s + (0.817 − 1.41i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.107 + 0.994i$
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: \((2,\ 588,\ (\ :1/2),\ -0.107 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06610 - 1.18797i\)
\(L(\frac12)\) \(\approx\) \(1.06610 - 1.18797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.121 + 1.40i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-2.88 - 1.66i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.810 + 0.468i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.87iT - 13T^{2} \)
17 \( 1 + (-4.51 + 2.60i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.56 + 6.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.810 - 0.468i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.561 - 0.972i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.46iT - 41T^{2} \)
43 \( 1 - 9.06iT - 43T^{2} \)
47 \( 1 + (-3.12 + 5.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.12 + 10.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.15 - 2.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.47 - 5.47i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.86iT - 71T^{2} \)
73 \( 1 + (5.77 - 3.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.07 - 1.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (-1.26 - 0.731i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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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

−10.49422838870042573580933239969, −9.765353240046489164675460374964, −9.114992881214086528500275349814, −7.80777880985281437228267465145, −6.73681659623418970376150081470, −5.69727377691620483369484775404, −5.02334574608318719732407276728, −3.25598279974576377440601097628, −2.46281549435720491040014640893, −1.11527948851283771903034948678, 1.44581367382757417274825573792, 3.58764727831791067831629083749, 4.69148274007942379267168047149, 5.70078494656625749460590373866, 5.97357912644756105380164276560, 7.29323014646653066623717644764, 8.356885240579617642424284450190, 9.279336463159618417408110271278, 9.749461023321388285782676322565, 10.54282348723236422209607588050

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