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Properties

Label 2-570-285.47-c1-0-29
Degree $2$
Conductor $570$
Sign $0.977 + 0.212i$
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  + (0.819 + 0.573i)2-s + (−1.06 − 1.36i)3-s + (0.342 + 0.939i)4-s + (2.13 − 0.651i)5-s + (−0.0861 − 1.72i)6-s + (1.82 − 0.489i)7-s + (−0.258 + 0.965i)8-s + (−0.740 + 2.90i)9-s + (2.12 + 0.693i)10-s + (−0.702 + 0.405i)11-s + (0.921 − 1.46i)12-s + (3.10 − 0.272i)13-s + (1.77 + 0.646i)14-s + (−3.16 − 2.23i)15-s + (−0.766 + 0.642i)16-s + (0.790 + 0.553i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (−0.613 − 0.789i)3-s + (0.171 + 0.469i)4-s + (0.956 − 0.291i)5-s + (−0.0351 − 0.706i)6-s + (0.690 − 0.184i)7-s + (−0.0915 + 0.341i)8-s + (−0.246 + 0.969i)9-s + (0.672 + 0.219i)10-s + (−0.211 + 0.122i)11-s + (0.266 − 0.423i)12-s + (0.862 − 0.0754i)13-s + (0.474 + 0.172i)14-s + (−0.816 − 0.576i)15-s + (−0.191 + 0.160i)16-s + (0.191 + 0.134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.977 + 0.212i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.977 + 0.212i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06312 - 0.221680i\)
\(L(\frac12)\) \(\approx\) \(2.06312 - 0.221680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 0.573i)T \)
3 \( 1 + (1.06 + 1.36i)T \)
5 \( 1 + (-2.13 + 0.651i)T \)
19 \( 1 + (1.15 + 4.20i)T \)
good7 \( 1 + (-1.82 + 0.489i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.702 - 0.405i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.10 + 0.272i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (-0.790 - 0.553i)T + (5.81 + 15.9i)T^{2} \)
23 \( 1 + (-0.114 - 0.0535i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (0.520 - 2.95i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.93 + 6.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.78 - 1.78i)T + 37iT^{2} \)
41 \( 1 + (-0.0602 - 0.0717i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (5.14 - 2.39i)T + (27.6 - 32.9i)T^{2} \)
47 \( 1 + (-4.18 - 5.98i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (3.97 + 1.85i)T + (34.0 + 40.6i)T^{2} \)
59 \( 1 + (1.44 + 8.17i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-8.78 + 3.19i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (5.34 - 3.73i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (3.39 - 9.33i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (0.988 - 11.3i)T + (-71.8 - 12.6i)T^{2} \)
79 \( 1 + (3.69 + 4.40i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (8.35 - 2.23i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (10.0 + 8.43i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (4.80 - 6.85i)T + (-33.1 - 91.1i)T^{2} \)
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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

−11.03691056993889375923431578195, −9.939711038343161064587146231441, −8.656756252411292447975215108684, −7.925284006533799016409113207752, −6.85581305680159942208712903283, −6.10416037980908952940514818196, −5.29961181574682618801735196499, −4.45532808757606026925297562432, −2.61942669495890022624666721609, −1.34273839623021978117176651647, 1.54532223789938753486168612916, 3.04565254969853471245304312231, 4.19654999366692698400232647672, 5.26190525842884137256129325339, 5.86728124124737671890536989191, 6.73725956471297016183273045744, 8.331113934801917299700616466817, 9.284272106798420861261324968877, 10.27352404061463117140942338762, 10.65325175968453726137376122139

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