Properties

Label 2-570-285.29-c1-0-24
Degree $2$
Conductor $570$
Sign $0.904 + 0.427i$
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  + (0.984 − 0.173i)2-s + (1.03 − 1.39i)3-s + (0.939 − 0.342i)4-s + (−0.272 + 2.21i)5-s + (0.775 − 1.54i)6-s + (1.48 + 0.856i)7-s + (0.866 − 0.5i)8-s + (−0.867 − 2.87i)9-s + (0.117 + 2.23i)10-s + (0.232 − 0.134i)11-s + (0.494 − 1.65i)12-s + (2.84 + 2.38i)13-s + (1.61 + 0.586i)14-s + (2.80 + 2.67i)15-s + (0.766 − 0.642i)16-s + (−0.267 − 1.51i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (0.596 − 0.802i)3-s + (0.469 − 0.171i)4-s + (−0.121 + 0.992i)5-s + (0.316 − 0.632i)6-s + (0.560 + 0.323i)7-s + (0.306 − 0.176i)8-s + (−0.289 − 0.957i)9-s + (0.0370 + 0.706i)10-s + (0.0702 − 0.0405i)11-s + (0.142 − 0.479i)12-s + (0.787 + 0.661i)13-s + (0.430 + 0.156i)14-s + (0.724 + 0.689i)15-s + (0.191 − 0.160i)16-s + (−0.0648 − 0.367i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.904 + 0.427i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.904 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69878 - 0.605925i\)
\(L(\frac12)\) \(\approx\) \(2.69878 - 0.605925i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (-1.03 + 1.39i)T \)
5 \( 1 + (0.272 - 2.21i)T \)
19 \( 1 + (-2.37 + 3.65i)T \)
good7 \( 1 + (-1.48 - 0.856i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.232 + 0.134i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.84 - 2.38i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.267 + 1.51i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.801 + 0.291i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.356 - 2.02i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.244 - 0.141i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.38T + 37T^{2} \)
41 \( 1 + (1.94 - 1.62i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.01 - 5.53i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.42 - 8.10i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.09 + 2.99i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (2.44 + 13.8i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (1.11 - 0.404i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.393 + 2.23i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.24 - 0.817i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.67 - 5.57i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-3.38 - 4.03i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (7.43 - 12.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (10.6 + 8.94i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.04 + 5.92i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.21351459501363025923382651589, −9.806248138151993971248081435988, −8.790962452674448545549572200853, −7.84648278102079129359839576649, −6.89771023287501139009677848658, −6.37531431815580950976149396736, −5.08767578503084527945286274632, −3.68820162612859521359995978828, −2.81743138130143485363130999255, −1.67290014447168631934578030555, 1.67655695772034906120459389023, 3.38113694891607310983192866725, 4.14152146452316762835345886757, 5.09487236926145977395213424517, 5.80931751096270524458435882124, 7.42300174586311323067178449193, 8.256130069415658023169650756941, 8.840028480547541742470352950713, 10.03458962404719373910388455473, 10.74960178127410728363937512208

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