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Properties

Label 2-570-285.83-c1-0-12
Degree $2$
Conductor $570$
Sign $-0.355 - 0.934i$
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  + (−0.258 + 0.965i)2-s + (1.72 + 0.158i)3-s + (−0.866 − 0.499i)4-s + (−1.67 − 1.48i)5-s + (−0.599 + 1.62i)6-s + (−2.12 + 2.12i)7-s + (0.707 − 0.707i)8-s + (2.94 + 0.548i)9-s + (1.86 − 1.23i)10-s + 5.82i·11-s + (−1.41 − 0.999i)12-s + (2.73 − 0.732i)13-s + (−1.5 − 2.59i)14-s + (−2.64 − 2.82i)15-s + (0.500 + 0.866i)16-s + (−1.61 + 6.02i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.995 + 0.0917i)3-s + (−0.433 − 0.249i)4-s + (−0.748 − 0.663i)5-s + (−0.244 + 0.663i)6-s + (−0.801 + 0.801i)7-s + (0.249 − 0.249i)8-s + (0.983 + 0.182i)9-s + (0.590 − 0.389i)10-s + 1.75i·11-s + (−0.408 − 0.288i)12-s + (0.757 − 0.203i)13-s + (−0.400 − 0.694i)14-s + (−0.684 − 0.729i)15-s + (0.125 + 0.216i)16-s + (−0.391 + 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.355 - 0.934i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.355 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.774658 + 1.12386i\)
\(L(\frac12)\) \(\approx\) \(0.774658 + 1.12386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-1.72 - 0.158i)T \)
5 \( 1 + (1.67 + 1.48i)T \)
19 \( 1 + (-4.33 + 0.5i)T \)
good7 \( 1 + (2.12 - 2.12i)T - 7iT^{2} \)
11 \( 1 - 5.82iT - 11T^{2} \)
13 \( 1 + (-2.73 + 0.732i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.61 - 6.02i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (0.776 + 2.89i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.41 - 7.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + (-2.12 + 2.12i)T - 37iT^{2} \)
41 \( 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.09 + 4.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-10.1 + 2.71i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.93 + 7.23i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.12 + 5.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.46 + 5.46i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.6 - 6.70i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.00 - 7.49i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.08 + 5.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3 + 3i)T - 83iT^{2} \)
89 \( 1 + (3.08 - 5.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.40 + 0.643i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.74503091737723960220820286300, −9.721433388919155245427960213812, −9.064221309004104481735627040882, −8.449555499228544764510597320654, −7.53816050585118443209110023505, −6.77612826542098814828164064438, −5.46104647305219259711159643968, −4.34793364933986064805913305729, −3.46399984253765825144869972949, −1.80781441363620314061739509994, 0.77620951638236886601840778658, 2.82001137644116429215567951312, 3.42891902906938218913035773438, 4.16407841422667702518966751780, 6.03155528906033273756823180451, 7.23529209018696132545333500829, 7.80196985815174298294940766356, 8.867887153922733309867154327803, 9.542553051019438192637632762496, 10.50401308027178355219508178708

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