Properties

Label 2-570-15.2-c1-0-5
Degree $2$
Conductor $570$
Sign $-0.247 - 0.968i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
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  + (0.707 − 0.707i)2-s + (0.336 + 1.69i)3-s − 1.00i·4-s + (−2.14 + 0.630i)5-s + (1.43 + 0.963i)6-s + (0.804 + 0.804i)7-s + (−0.707 − 0.707i)8-s + (−2.77 + 1.14i)9-s + (−1.07 + 1.96i)10-s + 3.15i·11-s + (1.69 − 0.336i)12-s + (−2.72 + 2.72i)13-s + 1.13·14-s + (−1.79 − 3.43i)15-s − 1.00·16-s + (−2.04 + 2.04i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.194 + 0.980i)3-s − 0.500i·4-s + (−0.959 + 0.282i)5-s + (0.587 + 0.393i)6-s + (0.304 + 0.304i)7-s + (−0.250 − 0.250i)8-s + (−0.924 + 0.381i)9-s + (−0.338 + 0.620i)10-s + 0.951i·11-s + (0.490 − 0.0972i)12-s + (−0.757 + 0.757i)13-s + 0.304·14-s + (−0.463 − 0.886i)15-s − 0.250·16-s + (−0.495 + 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.247 - 0.968i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.247 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.779777 + 1.00382i\)
\(L(\frac12)\) \(\approx\) \(0.779777 + 1.00382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.336 - 1.69i)T \)
5 \( 1 + (2.14 - 0.630i)T \)
19 \( 1 + iT \)
good7 \( 1 + (-0.804 - 0.804i)T + 7iT^{2} \)
11 \( 1 - 3.15iT - 11T^{2} \)
13 \( 1 + (2.72 - 2.72i)T - 13iT^{2} \)
17 \( 1 + (2.04 - 2.04i)T - 17iT^{2} \)
23 \( 1 + (-1.82 - 1.82i)T + 23iT^{2} \)
29 \( 1 + 5.49T + 29T^{2} \)
31 \( 1 - 5.07T + 31T^{2} \)
37 \( 1 + (-7.25 - 7.25i)T + 37iT^{2} \)
41 \( 1 - 6.31iT - 41T^{2} \)
43 \( 1 + (-0.643 + 0.643i)T - 43iT^{2} \)
47 \( 1 + (3.22 - 3.22i)T - 47iT^{2} \)
53 \( 1 + (3.16 + 3.16i)T + 53iT^{2} \)
59 \( 1 + 1.26T + 59T^{2} \)
61 \( 1 - 2.70T + 61T^{2} \)
67 \( 1 + (-3.86 - 3.86i)T + 67iT^{2} \)
71 \( 1 + 0.944iT - 71T^{2} \)
73 \( 1 + (-3.61 + 3.61i)T - 73iT^{2} \)
79 \( 1 + 15.0iT - 79T^{2} \)
83 \( 1 + (-5.54 - 5.54i)T + 83iT^{2} \)
89 \( 1 - 3.27T + 89T^{2} \)
97 \( 1 + (-2.89 - 2.89i)T + 97iT^{2} \)
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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

−11.28205076434552055105687496199, −10.12925965977455342044147293811, −9.495097826760550540926123608793, −8.487639796033234412541526573947, −7.48470612524232131720789001029, −6.34307659516511588621238647082, −4.83874502841331556069771237372, −4.49269905692200219771656610123, −3.37394529543075827134585165766, −2.23055743137532997859313078927, 0.58736188327762825954029782247, 2.64533563463535305540161979022, 3.74257108768722715186722192072, 4.95801046644584634803828400879, 5.95333456973950136763428679225, 7.06230371828256306452897621414, 7.71944585660910237993935832825, 8.327010034053281315106518716934, 9.239680481347022247594821560506, 10.89860940495308400695878860136

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