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Properties

Label 2-588-28.3-c1-0-14
Degree $2$
Conductor $588$
Sign $0.324 + 0.945i$
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  + (−1.22 − 0.709i)2-s + (−0.5 + 0.866i)3-s + (0.994 + 1.73i)4-s + (−3.15 + 1.82i)5-s + (1.22 − 0.705i)6-s + (0.0143 − 2.82i)8-s + (−0.499 − 0.866i)9-s + (5.15 + 0.00873i)10-s + (−5.25 − 3.03i)11-s + (−1.99 − 0.00677i)12-s + 0.483i·13-s − 3.64i·15-s + (−2.02 + 3.45i)16-s + (2.21 + 1.27i)17-s + (−0.00239 + 1.41i)18-s + (0.609 + 1.05i)19-s + ⋯
L(s)  = 1  + (−0.865 − 0.501i)2-s + (−0.288 + 0.499i)3-s + (0.497 + 0.867i)4-s + (−1.41 + 0.815i)5-s + (0.500 − 0.287i)6-s + (0.00508 − 0.999i)8-s + (−0.166 − 0.288i)9-s + (1.63 + 0.00276i)10-s + (−1.58 − 0.915i)11-s + (−0.577 − 0.00195i)12-s + 0.134i·13-s − 0.941i·15-s + (−0.505 + 0.862i)16-s + (0.536 + 0.309i)17-s + (−0.000564 + 0.333i)18-s + (0.139 + 0.242i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.324 + 0.945i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.324 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303068 - 0.216417i\)
\(L(\frac12)\) \(\approx\) \(0.303068 - 0.216417i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.709i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (3.15 - 1.82i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.25 + 3.03i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.483iT - 13T^{2} \)
17 \( 1 + (-2.21 - 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.609 - 1.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.00 + 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.21T + 29T^{2} \)
31 \( 1 + (-3.15 + 5.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.595 + 1.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.59iT - 41T^{2} \)
43 \( 1 + 3.51iT - 43T^{2} \)
47 \( 1 + (5.83 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.31 - 2.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.580 - 1.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.180 + 0.104i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.53 + 0.888i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.13iT - 71T^{2} \)
73 \( 1 + (5.23 + 3.02i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.75 + 1.58i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.49T + 83T^{2} \)
89 \( 1 + (-10.6 + 6.14i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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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

−10.51917637736649299877690576705, −10.03334965608532458976893212184, −8.568571962315041841135999170090, −8.059876674558342563479726169279, −7.26782416982980826518041055615, −6.15835328549033787456000068246, −4.65403231972295305011580153004, −3.44688096443407817007826832727, −2.81644839185784195556266703540, −0.36120995962245422140598367407, 1.02880289298671138792653785924, 2.81316484572393671857571177496, 4.74503234516177856677981106771, 5.25075470374671543179197104004, 6.72693373222360884788873230651, 7.60826082794105551831264468769, 7.987733169182223010735043656105, 8.829067965368561519446758373043, 9.987578128271344766842697727275, 10.78295963831599788308557223521

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