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Properties

Label 2-570-285.182-c1-0-21
Degree $2$
Conductor $570$
Sign $0.999 + 0.0239i$
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 + (−0.724 − 1.57i)3-s + (−0.866 + 0.499i)4-s + (1.67 − 1.48i)5-s + (1.33 − 1.10i)6-s + (2.12 + 2.12i)7-s + (−0.707 − 0.707i)8-s + (−1.94 + 2.28i)9-s + (1.86 + 1.23i)10-s − 0.171i·11-s + (1.41 + 1.00i)12-s + (2.73 + 0.732i)13-s + (−1.5 + 2.59i)14-s + (−3.54 − 1.55i)15-s + (0.500 − 0.866i)16-s + (−0.580 − 2.16i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.418 − 0.908i)3-s + (−0.433 + 0.249i)4-s + (0.748 − 0.663i)5-s + (0.543 − 0.452i)6-s + (0.801 + 0.801i)7-s + (−0.249 − 0.249i)8-s + (−0.649 + 0.760i)9-s + (0.590 + 0.389i)10-s − 0.0517i·11-s + (0.408 + 0.288i)12-s + (0.757 + 0.203i)13-s + (−0.400 + 0.694i)14-s + (−0.915 − 0.401i)15-s + (0.125 − 0.216i)16-s + (−0.140 − 0.525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.999 + 0.0239i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.999 + 0.0239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66074 - 0.0199001i\)
\(L(\frac12)\) \(\approx\) \(1.66074 - 0.0199001i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.724 + 1.57i)T \)
5 \( 1 + (-1.67 + 1.48i)T \)
19 \( 1 + (-4.33 - 0.5i)T \)
good7 \( 1 + (-2.12 - 2.12i)T + 7iT^{2} \)
11 \( 1 + 0.171iT - 11T^{2} \)
13 \( 1 + (-2.73 - 0.732i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.580 + 2.16i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (-0.776 + 2.89i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.58 + 2.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
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 \( 1 + (-6.26 - 1.67i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.45 - 9.16i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.82 + 4.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.12 + 1.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.46 - 5.46i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.16 + 5.29i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.20 - 15.6i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.61 + 3.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3 - 3i)T + 83iT^{2} \)
89 \( 1 + (5.91 + 10.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.9 - 3.74i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.93049229322806006001383370981, −9.596234800533181372522370146456, −8.665360535273070652747568088828, −8.128777538688053688971899713698, −7.03455881563039761067179971103, −6.01300841478260382018715801879, −5.46817143365874079189513114976, −4.57573895941544668667359279033, −2.56373910664505528914914099939, −1.21959934438064179411595442367, 1.37870596354219963343027031223, 3.06017642235294110050278247284, 3.98247904363451002408921513844, 5.06809875986798710503441083495, 5.87032494780977609358480109885, 7.00548871128166573509697044089, 8.321124770462314670833179112688, 9.361097073345192523376225176316, 10.14699230038955043372390179054, 10.75495380432161905076173220957

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