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Properties

Label 2-570-95.27-c1-0-10
Degree $2$
Conductor $570$
Sign $0.344 - 0.938i$
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 + (0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (2.23 − 0.154i)5-s + (0.499 + 0.866i)6-s + (3.62 + 3.62i)7-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (0.726 + 2.11i)10-s − 5.42·11-s + (−0.707 + 0.707i)12-s + (0.0914 − 0.341i)13-s + (−2.56 + 4.44i)14-s + (2.11 − 0.726i)15-s + (0.500 − 0.866i)16-s + (4.93 − 1.32i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.997 − 0.0689i)5-s + (0.204 + 0.353i)6-s + (1.37 + 1.37i)7-s + (−0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (0.229 + 0.668i)10-s − 1.63·11-s + (−0.204 + 0.204i)12-s + (0.0253 − 0.0946i)13-s + (−0.685 + 1.18i)14-s + (0.546 − 0.187i)15-s + (0.125 − 0.216i)16-s + (1.19 − 0.320i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.344 - 0.938i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.344 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86754 + 1.30354i\)
\(L(\frac12)\) \(\approx\) \(1.86754 + 1.30354i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-2.23 + 0.154i)T \)
19 \( 1 + (4.35 + 0.110i)T \)
good7 \( 1 + (-3.62 - 3.62i)T + 7iT^{2} \)
11 \( 1 + 5.42T + 11T^{2} \)
13 \( 1 + (-0.0914 + 0.341i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.93 + 1.32i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.154 - 0.0412i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.21 + 7.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.34iT - 31T^{2} \)
37 \( 1 + (3.07 - 3.07i)T - 37iT^{2} \)
41 \( 1 + (-6.19 - 3.57i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.749 + 2.79i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.26 - 4.71i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.55 + 5.81i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.56 + 4.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.49 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.54 + 1.21i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.2 + 6.49i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.65 - 6.16i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.63 + 8.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.61 + 8.61i)T - 83iT^{2} \)
89 \( 1 + (-0.358 - 0.621i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.377 + 1.41i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.78994193033894873521392140192, −9.827356092438952566595273139893, −8.923725515394052431545392833957, −8.141740521710042664093797851234, −7.64838220806743450525640480520, −6.11091304686111628179454500050, −5.42724750244261082229072138210, −4.74332717038816071984087470171, −2.83378082838365988278894152282, −1.95267695277506602432797818254, 1.39359360385916875988561890387, 2.44724898434565391623342429501, 3.77730801765126876876020693445, 4.86338799057725037314379572234, 5.59159793491707954605617017363, 7.23872481459104733586197184936, 7.969146799398413184009787774235, 8.875187130238301509579695726314, 10.07575903524200806193485268930, 10.57669430508332358446779988258

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