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Properties

Label 2-570-285.284-c1-0-4
Degree $2$
Conductor $570$
Sign $-0.332 - 0.942i$
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  − i·2-s + (−1.41 + i)3-s − 4-s + (1.73 + 1.41i)5-s + (1 + 1.41i)6-s + 2.44i·7-s + i·8-s + (1.00 − 2.82i)9-s + (1.41 − 1.73i)10-s + 1.41i·11-s + (1.41 − i)12-s − 4.24·13-s + 2.44·14-s + (−3.86 − 0.267i)15-s + 16-s − 6.92·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.816 + 0.577i)3-s − 0.5·4-s + (0.774 + 0.632i)5-s + (0.408 + 0.577i)6-s + 0.925i·7-s + 0.353i·8-s + (0.333 − 0.942i)9-s + (0.447 − 0.547i)10-s + 0.426i·11-s + (0.408 − 0.288i)12-s − 1.17·13-s + 0.654·14-s + (−0.997 − 0.0691i)15-s + 0.250·16-s − 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.332 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.332 - 0.942i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.332 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413303 + 0.584227i\)
\(L(\frac12)\) \(\approx\) \(0.413303 + 0.584227i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.41 - i)T \)
5 \( 1 + (-1.73 - 1.41i)T \)
19 \( 1 + (-4 + 1.73i)T \)
good7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 2.44T + 29T^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 7.34iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 4.89T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 4.89T + 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 2.44T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.99473243840534574957528127692, −10.15475585978894969327056097488, −9.528809673339358144590040580913, −8.872764621346993315758833640343, −7.21505323285910910383690117731, −6.29403660840370293367239834773, −5.29623806381723137280014706696, −4.57442960229952385474681392151, −3.04860567638519943019597844695, −2.00209811176355755722668978789, 0.42999619596698241405942853346, 2.04522298927654473501922214227, 4.21140715351182110625618874436, 5.09703688974612198193761688609, 5.92448122360297994972802729383, 6.83638481188482870632889438024, 7.54997388063697493413502827506, 8.544919733494520607725820688939, 9.681735766562226004695934830015, 10.31086181434782771490296946219

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