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Properties

Label 2-570-95.88-c1-0-14
Degree $2$
Conductor $570$
Sign $0.519 + 0.854i$
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 + (1.87 − 1.21i)5-s + (0.499 − 0.866i)6-s + (−0.219 + 0.219i)7-s + (−0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.690 − 2.12i)10-s + 6.33·11-s + (−0.707 − 0.707i)12-s + (0.962 + 3.59i)13-s + (0.155 + 0.268i)14-s + (2.12 − 0.690i)15-s + (0.500 + 0.866i)16-s + (−3.30 − 0.885i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.557 + 0.149i)3-s + (−0.433 − 0.249i)4-s + (0.838 − 0.544i)5-s + (0.204 − 0.353i)6-s + (−0.0828 + 0.0828i)7-s + (−0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.218 − 0.672i)10-s + 1.90·11-s + (−0.204 − 0.204i)12-s + (0.267 + 0.996i)13-s + (0.0414 + 0.0717i)14-s + (0.549 − 0.178i)15-s + (0.125 + 0.216i)16-s + (−0.801 − 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.519 + 0.854i$
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: \((2,\ 570,\ (\ :1/2),\ 0.519 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92098 - 1.08034i\)
\(L(\frac12)\) \(\approx\) \(1.92098 - 1.08034i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 + 1.21i)T \)
19 \( 1 + (3.97 + 1.79i)T \)
good7 \( 1 + (0.219 - 0.219i)T - 7iT^{2} \)
11 \( 1 - 6.33T + 11T^{2} \)
13 \( 1 + (-0.962 - 3.59i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.30 + 0.885i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (2.24 - 0.601i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.42 + 5.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.71iT - 31T^{2} \)
37 \( 1 + (0.817 + 0.817i)T + 37iT^{2} \)
41 \( 1 + (8.81 - 5.08i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.26 + 8.44i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.73 - 10.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.06 - 3.96i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.367 + 0.636i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.17 + 3.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.98 - 1.33i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.14 - 0.663i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.24 - 4.65i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.21 + 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.190 + 0.190i)T + 83iT^{2} \)
89 \( 1 + (5.18 - 8.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.26 - 15.9i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.54154337679969125144555322760, −9.421858832657940029365560490562, −9.199999431575972621038024580223, −8.398595674211101762111542072434, −6.72922851806779701761994443631, −6.08431010587121939581951239471, −4.52590746351196295811984570615, −4.02815988588267544394877025753, −2.43152384874359198690873488064, −1.45160739064990564611878942841, 1.68359193161362257447705340928, 3.21706908070456223507278515425, 4.17728286553586416754678013054, 5.58975109015255129762745655089, 6.60331754609796230124051841489, 6.91638200651609054594221133110, 8.411381263512589903024119096520, 8.870413255401084260574814157355, 9.907162563283045290981563706474, 10.64517934326735189563188482972

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