Properties

Label 2-570-285.269-c1-0-0
Degree $2$
Conductor $570$
Sign $-0.992 - 0.118i$
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  + (0.642 − 0.766i)2-s + (0.170 + 1.72i)3-s + (−0.173 − 0.984i)4-s + (−2.23 − 0.0827i)5-s + (1.43 + 0.977i)6-s + (−1.14 + 0.658i)7-s + (−0.866 − 0.500i)8-s + (−2.94 + 0.587i)9-s + (−1.49 + 1.65i)10-s + (−2.83 − 1.63i)11-s + (1.66 − 0.467i)12-s + (0.592 + 0.215i)13-s + (−0.228 + 1.29i)14-s + (−0.238 − 3.86i)15-s + (−0.939 + 0.342i)16-s + (−3.12 − 2.61i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.0984 + 0.995i)3-s + (−0.0868 − 0.492i)4-s + (−0.999 − 0.0370i)5-s + (0.583 + 0.398i)6-s + (−0.431 + 0.248i)7-s + (−0.306 − 0.176i)8-s + (−0.980 + 0.195i)9-s + (−0.474 + 0.524i)10-s + (−0.854 − 0.493i)11-s + (0.481 − 0.134i)12-s + (0.164 + 0.0598i)13-s + (−0.0611 + 0.346i)14-s + (−0.0615 − 0.998i)15-s + (−0.234 + 0.0855i)16-s + (−0.757 − 0.635i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.992 - 0.118i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.992 - 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00465229 + 0.0782986i\)
\(L(\frac12)\) \(\approx\) \(0.00465229 + 0.0782986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (-0.170 - 1.72i)T \)
5 \( 1 + (2.23 + 0.0827i)T \)
19 \( 1 + (4.29 - 0.729i)T \)
good7 \( 1 + (1.14 - 0.658i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.83 + 1.63i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.592 - 0.215i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.12 + 2.61i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.00940 + 0.0533i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (6.16 - 5.17i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.01 + 0.586i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.17T + 37T^{2} \)
41 \( 1 + (4.99 - 1.81i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-5.75 - 1.01i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.96 + 2.48i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (5.06 - 0.893i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (4.27 + 3.58i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.166 - 0.946i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.52 - 1.28i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.158 + 0.899i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-3.00 - 8.24i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (4.62 + 12.7i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-3.82 - 6.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-13.5 - 4.91i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-11.3 - 9.53i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.97378247512077153312026684743, −10.65733721000750412656303618486, −9.438376409019225350485986888678, −8.751742209791293158855058266512, −7.79676593062221982738485818012, −6.42068506972298150997130134181, −5.30207650862213836891777274674, −4.43456815509486491645131249680, −3.52057996134441075685446255754, −2.61925983541828976625051954628, 0.03513930379635158079281794658, 2.31773626360132642423118222933, 3.59308762798050880918561132675, 4.63304848973661374188902672008, 5.95880925681218919930270789185, 6.77075163342929590779302691026, 7.58206211573421689319437194253, 8.173786380567064722234366987929, 9.079192691250913032953179884864, 10.55895842281821680711039684274

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