Properties

Label 2-570-19.6-c1-0-1
Degree $2$
Conductor $570$
Sign $-0.765 - 0.643i$
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  + (−0.766 + 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.939 + 0.342i)6-s + (−2.35 + 4.08i)7-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (0.766 + 1.32i)11-s + (0.499 − 0.866i)12-s + (−2.24 + 0.817i)13-s + (−0.819 − 4.64i)14-s + (0.173 − 0.984i)15-s + (−0.939 − 0.342i)16-s + (−0.184 + 0.155i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.542 + 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.0776 − 0.440i)5-s + (−0.383 + 0.139i)6-s + (−0.891 + 1.54i)7-s + (0.176 + 0.306i)8-s + (0.255 + 0.214i)9-s + (0.242 + 0.203i)10-s + (0.230 + 0.400i)11-s + (0.144 − 0.250i)12-s + (−0.622 + 0.226i)13-s + (−0.218 − 1.24i)14-s + (0.0448 − 0.254i)15-s + (−0.234 − 0.0855i)16-s + (−0.0448 + 0.0376i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.765 - 0.643i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.765 - 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.291918 + 0.801224i\)
\(L(\frac12)\) \(\approx\) \(0.291918 + 0.801224i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (2.23 + 3.74i)T \)
good7 \( 1 + (2.35 - 4.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.766 - 1.32i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.24 - 0.817i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.184 - 0.155i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (1.16 - 6.60i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.34 - 1.13i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (3.41 - 5.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.39T + 37T^{2} \)
41 \( 1 + (-11.2 - 4.09i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.29 - 7.32i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-4.17 - 3.50i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-1.04 + 5.93i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (7.24 - 6.07i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.453 + 2.57i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (3.16 + 2.65i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.26 + 7.19i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-4.10 - 1.49i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-9.06 - 3.29i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-4.94 + 8.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.75 + 1.72i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (4.41 - 3.70i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.92381989977822781077067778855, −9.688565088516093150512581613638, −9.252923665820799899638884477327, −8.709421763141098411737421898751, −7.62582216274927240566421381845, −6.65567150672832783796659227889, −5.65347400000297767898731927471, −4.69970912279363371544602658353, −3.14626901249478427120138495366, −1.97554648400314666676274647494, 0.51954211405115910775916284956, 2.31631145562151188860107344220, 3.52749990158142682845281874845, 4.18934181253212892707726585257, 6.14995940272950792349646338866, 7.08676442265336925116033575125, 7.66434931050191118104202950028, 8.704644346351172766719479802488, 9.682664542446652750534379822660, 10.42049263417773601039692936946

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