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Properties

Label 2-570-95.88-c1-0-6
Degree $2$
Conductor $570$
Sign $0.129 - 0.991i$
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 + 0.109i)5-s + (0.499 − 0.866i)6-s + (−0.613 + 0.613i)7-s + (0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.683 + 2.12i)10-s − 3.43·11-s + (0.707 + 0.707i)12-s + (0.543 + 2.02i)13-s + (−0.434 − 0.751i)14-s + (−2.12 − 0.683i)15-s + (0.500 + 0.866i)16-s + (5.79 + 1.55i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 − 0.249i)4-s + (0.998 + 0.0489i)5-s + (0.204 − 0.353i)6-s + (−0.232 + 0.232i)7-s + (0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.216 + 0.673i)10-s − 1.03·11-s + (0.204 + 0.204i)12-s + (0.150 + 0.562i)13-s + (−0.116 − 0.200i)14-s + (−0.549 − 0.176i)15-s + (0.125 + 0.216i)16-s + (1.40 + 0.376i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882214 + 0.774436i\)
\(L(\frac12)\) \(\approx\) \(0.882214 + 0.774436i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.109i)T \)
19 \( 1 + (-3.87 - 2.00i)T \)
good7 \( 1 + (0.613 - 0.613i)T - 7iT^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 + (-0.543 - 2.02i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-5.79 - 1.55i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (-2.31 + 0.621i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.59 - 7.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.09iT - 31T^{2} \)
37 \( 1 + (-7.31 - 7.31i)T + 37iT^{2} \)
41 \( 1 + (6.91 - 3.99i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.968 - 3.61i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.89 - 7.08i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.887 + 3.31i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.68 - 6.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.34 + 5.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.18 + 1.38i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.85 - 3.38i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.07 + 11.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.09 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.509 - 0.509i)T + 83iT^{2} \)
89 \( 1 + (-6.81 + 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.874 + 3.26i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.73495142206472512277879990229, −9.905561124477596596218066085649, −9.362214858649881120742300986089, −8.116646539378126863092529136524, −7.31766977250925483918306474089, −6.21603723178728061383154799807, −5.64285813605056473814810826348, −4.82769545314140582466135684677, −3.08920053303855974663535535631, −1.42278567051605193241184149337, 0.864231256633635259112675728647, 2.47671007147439845292972420282, 3.59314350272571368520346068441, 5.27981054607541660953601527182, 5.47385988667680265766914592639, 6.95965880289530189335927932124, 7.919112987946015587087853538342, 9.122395952456723965672393243502, 9.998148990026682602995933768783, 10.32770014429717787770265112486

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