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Properties

Label 2-570-95.12-c1-0-15
Degree $2$
Conductor $570$
Sign $-0.513 + 0.858i$
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  + (−0.965 − 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.0684 − 2.23i)5-s + (−0.499 + 0.866i)6-s + (−0.515 − 0.515i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.512 + 2.17i)10-s + 6.29·11-s + (0.707 − 0.707i)12-s + (0.806 − 0.216i)13-s + (0.364 + 0.631i)14-s + (−2.17 − 0.512i)15-s + (0.500 + 0.866i)16-s + (0.362 − 1.35i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + (−0.0306 − 0.999i)5-s + (−0.204 + 0.353i)6-s + (−0.194 − 0.194i)7-s + (−0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.162 + 0.688i)10-s + 1.89·11-s + (0.204 − 0.204i)12-s + (0.223 − 0.0599i)13-s + (0.0974 + 0.168i)14-s + (−0.561 − 0.132i)15-s + (0.125 + 0.216i)16-s + (0.0878 − 0.327i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.513 + 0.858i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.513 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.524586 - 0.924970i\)
\(L(\frac12)\) \(\approx\) \(0.524586 - 0.924970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.0684 + 2.23i)T \)
19 \( 1 + (3.82 + 2.09i)T \)
good7 \( 1 + (0.515 + 0.515i)T + 7iT^{2} \)
11 \( 1 - 6.29T + 11T^{2} \)
13 \( 1 + (-0.806 + 0.216i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.362 + 1.35i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (1.53 + 5.74i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.900 + 1.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.93iT - 31T^{2} \)
37 \( 1 + (0.856 - 0.856i)T - 37iT^{2} \)
41 \( 1 + (-4.31 + 2.49i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.63 + 1.77i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.26 + 1.14i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (13.0 - 3.49i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.106 + 0.183i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.48 - 2.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.23 + 8.34i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-6.29 + 3.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.15 - 1.64i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.71 - 2.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.52 + 7.52i)T - 83iT^{2} \)
89 \( 1 + (3.15 - 5.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.93 - 1.05i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.35393180422951244352664279992, −9.228908119350523948079389531318, −8.841140178421418226297344650315, −8.014977236919708922444428657784, −6.80824933476619836258481384595, −6.25574270491606424712107520432, −4.68431938571047550023243901331, −3.59019642947452782259603791964, −1.93507316468161089628179158144, −0.77837921269711390297367286108, 1.80828406629770420744147226278, 3.32093507320963614624542464082, 4.17740083420531996868113172646, 5.96323656493071200132902500630, 6.45211463486941357972636889877, 7.52570559880306713701209258543, 8.508844845536504180141620516776, 9.467904697723931688897881875061, 9.894759654474987929865959896411, 11.04542051459657495585129558437

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