↑

Properties

Label 2-570-95.88-c1-0-11
Degree $2$
Conductor $570$
Sign $0.537 + 0.843i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 − 0.499i)4-s + (2.08 + 0.803i)5-s + (−0.499 + 0.866i)6-s + (0.475 − 0.475i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (1.31 − 1.80i)10-s + 1.62·11-s + (0.707 + 0.707i)12-s + (0.323 + 1.20i)13-s + (−0.336 − 0.582i)14-s + (−1.80 − 1.31i)15-s + (0.500 + 0.866i)16-s + (2.66 + 0.715i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 − 0.249i)4-s + (0.933 + 0.359i)5-s + (−0.204 + 0.353i)6-s + (0.179 − 0.179i)7-s + (−0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (0.416 − 0.571i)10-s + 0.490·11-s + (0.204 + 0.204i)12-s + (0.0898 + 0.335i)13-s + (−0.0898 − 0.155i)14-s + (−0.466 − 0.339i)15-s + (0.125 + 0.216i)16-s + (0.647 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.537 + 0.843i$
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: \((2,\ 570,\ (\ :1/2),\ 0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40459 - 0.770862i\)
\(L(\frac12)\) \(\approx\) \(1.40459 - 0.770862i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.08 - 0.803i)T \)
19 \( 1 + (-3.74 + 2.23i)T \)
good7 \( 1 + (-0.475 + 0.475i)T - 7iT^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 + (-0.323 - 1.20i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-2.66 - 0.715i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (1.29 - 0.346i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.85 + 4.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.96iT - 31T^{2} \)
37 \( 1 + (1.34 + 1.34i)T + 37iT^{2} \)
41 \( 1 + (-3.11 + 1.80i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.07 - 4.02i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.70 + 10.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.297 - 1.10i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.21 + 5.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.415 - 0.719i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.62 - 1.23i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-6.55 + 3.78i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.10 - 11.5i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.50 - 6.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.66 - 4.66i)T + 83iT^{2} \)
89 \( 1 + (-0.342 + 0.593i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.83 - 6.83i)T + (-84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.67632728417810999867252616619, −9.829545240379986852525568591248, −9.263564605484074267454612007549, −7.918979254025749031138801259573, −6.76963928721928241836209415876, −5.92037690502834630021867232233, −5.04023292889407329114477084000, −3.83769963099399578056992518242, −2.47006770830259023108734408286, −1.20893562339177576822504330201, 1.34102342813916424584412728811, 3.23826307758116137746167006079, 4.67866985716927995431646420863, 5.44896007379123923460892135691, 6.15071838999653393713366252112, 7.10489716133415519266778454696, 8.216776010043901515739085985206, 9.147681623715722820810747901699, 9.903365821498149322475921679448, 10.72803019073046544669424977230

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