Properties

Label 2-570-285.59-c1-0-7
Degree $2$
Conductor $570$
Sign $-0.317 - 0.948i$
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 + (1.27 + 1.16i)3-s + (0.939 + 0.342i)4-s + (2.18 − 0.491i)5-s + (−1.05 − 1.37i)6-s + (−3.69 + 2.13i)7-s + (−0.866 − 0.5i)8-s + (0.268 + 2.98i)9-s + (−2.23 + 0.105i)10-s + (−2.36 − 1.36i)11-s + (0.801 + 1.53i)12-s + (−2.74 + 2.30i)13-s + (4.01 − 1.46i)14-s + (3.36 + 1.92i)15-s + (0.766 + 0.642i)16-s + (−1.27 + 7.23i)17-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (0.738 + 0.674i)3-s + (0.469 + 0.171i)4-s + (0.975 − 0.219i)5-s + (−0.431 − 0.560i)6-s + (−1.39 + 0.806i)7-s + (−0.306 − 0.176i)8-s + (0.0893 + 0.996i)9-s + (−0.706 + 0.0332i)10-s + (−0.712 − 0.411i)11-s + (0.231 + 0.443i)12-s + (−0.762 + 0.639i)13-s + (1.07 − 0.390i)14-s + (0.868 + 0.496i)15-s + (0.191 + 0.160i)16-s + (−0.309 + 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646023 + 0.897746i\)
\(L(\frac12)\) \(\approx\) \(0.646023 + 0.897746i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.27 - 1.16i)T \)
5 \( 1 + (-2.18 + 0.491i)T \)
19 \( 1 + (-0.726 - 4.29i)T \)
good7 \( 1 + (3.69 - 2.13i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.36 + 1.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.74 - 2.30i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.27 - 7.23i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.504 + 0.183i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.17 + 6.65i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-6.25 + 3.61i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.30T + 37T^{2} \)
41 \( 1 + (-4.84 - 4.06i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.172 + 0.473i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.399 + 2.26i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-0.105 + 0.291i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.70 - 9.66i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-1.45 - 0.530i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.976 - 5.53i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.79 - 1.01i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-8.13 + 9.69i)T + (-12.6 - 71.8i)T^{2} \)
79 \( 1 + (9.01 - 10.7i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (7.26 + 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.27 + 6.10i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (0.308 - 1.75i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.43129341820185551893196641573, −9.961616645220952825383607941659, −9.403244577657301687035656473317, −8.614000400643872809628757786315, −7.81224749626496175801437865125, −6.28442921856019153664213714091, −5.78036985460453891736833769402, −4.20302787511442226826513505080, −2.86321120988743430599635293042, −2.12537782392352943894042235151, 0.67682509119439261158326630343, 2.53720563891818501276915150229, 3.03465868619927314863537122604, 5.01880997459938805166047587443, 6.37073356382297139662729452477, 7.07072585236014909427270198088, 7.54359510033508820560224828245, 8.932425869836501688607089336726, 9.662877857533773643442611593660, 10.00635045707473534370601318383

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