Properties

Label 2-570-19.16-c1-0-5
Degree $2$
Conductor $570$
Sign $0.766 + 0.642i$
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 + (0.266 + 0.460i)7-s + (0.500 − 0.866i)8-s + (0.766 − 0.642i)9-s + (−0.766 + 0.642i)10-s + (−0.326 + 0.565i)11-s + (−0.499 − 0.866i)12-s + (0.439 + 0.160i)13-s + (0.0923 − 0.524i)14-s + (0.173 + 0.984i)15-s + (−0.939 + 0.342i)16-s + (0.439 + 0.368i)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 + (0.100 + 0.174i)7-s + (0.176 − 0.306i)8-s + (0.255 − 0.214i)9-s + (−0.242 + 0.203i)10-s + (−0.0983 + 0.170i)11-s + (−0.144 − 0.250i)12-s + (0.121 + 0.0443i)13-s + (0.0246 − 0.140i)14-s + (0.0448 + 0.254i)15-s + (−0.234 + 0.0855i)16-s + (0.106 + 0.0894i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895473 - 0.325594i\)
\(L(\frac12)\) \(\approx\) \(0.895473 - 0.325594i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.07 - 1.55i)T \)
good7 \( 1 + (-0.266 - 0.460i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.326 - 0.565i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.439 - 0.160i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.439 - 0.368i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.09 + 6.19i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-4.55 + 3.82i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.05 - 7.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.17T + 37T^{2} \)
41 \( 1 + (-9.99 + 3.63i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.578 + 3.28i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-3.75 + 3.15i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (1.86 + 10.5i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-7.07 - 5.93i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.503 + 2.85i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.88 - 6.61i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.17 - 6.67i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (0.247 - 0.0901i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-2.49 + 0.907i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (1.78 + 3.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.72 + 1.35i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-2.89 - 2.42i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.45295301125221404741013592098, −9.982834848975447946778750520534, −8.935168859175879057901491503070, −8.211436989972164418038831277958, −7.12545996505014054088659282945, −6.04709470795218191285457429648, −4.98587695659416134640466701268, −3.96266798603349577457080102312, −2.49273022674091842537828904015, −0.917200015963794142517730741304, 1.10957861463478056120659007418, 2.82587386145345647755846696205, 4.41084535118673877238889793271, 5.61723640639435822383588298476, 6.29461541071027269162351551178, 7.40487314964740248254239244468, 7.86733640996083451275907047053, 9.216793340351153519781004410842, 9.873317691904223564671856762654, 10.88616895869109935654937625539

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