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Properties

Label 2-570-95.27-c1-0-9
Degree $2$
Conductor $570$
Sign $0.872 + 0.488i$
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  + (−0.258 − 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (1.82 + 1.28i)5-s + (−0.499 − 0.866i)6-s + (0.555 + 0.555i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (0.770 − 2.09i)10-s − 1.49·11-s + (−0.707 + 0.707i)12-s + (0.362 − 1.35i)13-s + (0.392 − 0.680i)14-s + (2.09 + 0.770i)15-s + (0.500 − 0.866i)16-s + (4.28 − 1.14i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.817 + 0.575i)5-s + (−0.204 − 0.353i)6-s + (0.210 + 0.210i)7-s + (0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (0.243 − 0.663i)10-s − 0.452·11-s + (−0.204 + 0.204i)12-s + (0.100 − 0.374i)13-s + (0.105 − 0.181i)14-s + (0.542 + 0.198i)15-s + (0.125 − 0.216i)16-s + (1.03 − 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76396 - 0.460047i\)
\(L(\frac12)\) \(\approx\) \(1.76396 - 0.460047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-1.82 - 1.28i)T \)
19 \( 1 + (-2.08 - 3.82i)T \)
good7 \( 1 + (-0.555 - 0.555i)T + 7iT^{2} \)
11 \( 1 + 1.49T + 11T^{2} \)
13 \( 1 + (-0.362 + 1.35i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.28 + 1.14i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.867 - 0.232i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.46 - 7.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.39iT - 31T^{2} \)
37 \( 1 + (-1.66 + 1.66i)T - 37iT^{2} \)
41 \( 1 + (4.17 + 2.40i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.27 + 4.74i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.16 - 11.7i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.555 + 2.07i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.90 - 3.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.36 + 9.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.00251 + 0.000672i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.8 + 6.87i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.459 - 1.71i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.131 - 0.227i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.84 - 9.84i)T - 83iT^{2} \)
89 \( 1 + (6.69 + 11.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.86 - 6.96i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.45431272833699004473218235771, −9.923926827279289318371809313212, −9.103104192118322555903303604475, −8.079853818887292143086101491335, −7.32750942267515928697122999780, −6.00447616305376268503137580428, −5.07951391571137745601342655895, −3.50320581257367270100865473608, −2.69695759647840958400658111525, −1.49666035436154794203867756265, 1.33764548635465113581905141595, 2.92059296006088969655717082436, 4.46839352123277342890968439579, 5.24308022002042757315359401269, 6.28728674482563696102114383732, 7.32224266088377150554320726415, 8.281927284018794862552423609140, 8.900028349341376091179730820311, 9.870047687503744209076065199791, 10.33915536199191165360946856890

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