Properties

Label 2-570-285.29-c1-0-20
Degree $2$
Conductor $570$
Sign $0.249 + 0.968i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 + 0.173i)2-s + (−1.23 − 1.21i)3-s + (0.939 − 0.342i)4-s + (1.86 − 1.22i)5-s + (1.42 + 0.977i)6-s + (−1.22 − 0.709i)7-s + (−0.866 + 0.5i)8-s + (0.0679 + 2.99i)9-s + (−1.62 + 1.53i)10-s + (4.15 − 2.40i)11-s + (−1.57 − 0.714i)12-s + (4.38 + 3.68i)13-s + (1.33 + 0.485i)14-s + (−3.80 − 0.739i)15-s + (0.766 − 0.642i)16-s + (−0.869 − 4.92i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (−0.715 − 0.699i)3-s + (0.469 − 0.171i)4-s + (0.835 − 0.549i)5-s + (0.583 + 0.398i)6-s + (−0.464 − 0.268i)7-s + (−0.306 + 0.176i)8-s + (0.0226 + 0.999i)9-s + (−0.514 + 0.485i)10-s + (1.25 − 0.723i)11-s + (−0.455 − 0.206i)12-s + (1.21 + 1.02i)13-s + (0.356 + 0.129i)14-s + (−0.981 − 0.190i)15-s + (0.191 − 0.160i)16-s + (−0.210 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.249 + 0.968i$
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: \((2,\ 570,\ (\ :1/2),\ 0.249 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782329 - 0.606461i\)
\(L(\frac12)\) \(\approx\) \(0.782329 - 0.606461i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 + 1.21i)T \)
5 \( 1 + (-1.86 + 1.22i)T \)
19 \( 1 + (-3.92 - 1.90i)T \)
good7 \( 1 + (1.22 + 0.709i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.15 + 2.40i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.38 - 3.68i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.869 + 4.92i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (4.49 - 1.63i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.843 - 4.78i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (8.38 + 4.84i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.32T + 37T^{2} \)
41 \( 1 + (-3.61 + 3.03i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.09 + 8.49i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.195 - 1.10i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (0.0601 + 0.165i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.697 + 3.95i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.668 + 0.243i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.07 + 11.7i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (11.1 + 4.05i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.496 + 0.591i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-6.74 - 8.03i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.06 - 1.84i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.424 - 0.356i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-3.40 - 19.2i)T + (-91.1 + 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.60865848791208512014150120087, −9.312706754959804574385403370102, −9.147878231016841963910291479431, −7.83306668363657341430950339266, −6.79171165429924633056620557553, −6.18220949574267729474456706346, −5.40444845139389836510269284710, −3.80767077835900723373943328987, −1.91736782219899809968570048246, −0.895906477991350848486098176846, 1.41551650579555078329311989376, 3.10790752953363813327000629115, 4.17060775107373551351679614855, 5.95426547240632511799768948614, 6.05935042221045418289087297269, 7.20983163954897285872412298799, 8.603707788478692251825527725973, 9.487321417748244381826559223559, 9.939510116201404728934238632860, 10.82623645411479991266989236564

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