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Properties

Label 2-570-95.12-c1-0-14
Degree $2$
Conductor $570$
Sign $0.990 + 0.140i$
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 + (1.89 − 1.19i)5-s + (0.499 − 0.866i)6-s + (2.32 + 2.32i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (2.13 − 0.660i)10-s + 3.35·11-s + (0.707 − 0.707i)12-s + (−5.53 + 1.48i)13-s + (1.64 + 2.84i)14-s + (−0.660 − 2.13i)15-s + (0.500 + 0.866i)16-s + (−0.215 + 0.802i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + (0.846 − 0.532i)5-s + (0.204 − 0.353i)6-s + (0.878 + 0.878i)7-s + (0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.675 − 0.208i)10-s + 1.01·11-s + (0.204 − 0.204i)12-s + (−1.53 + 0.411i)13-s + (0.439 + 0.760i)14-s + (−0.170 − 0.551i)15-s + (0.125 + 0.216i)16-s + (−0.0521 + 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.990 + 0.140i$
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: \((2,\ 570,\ (\ :1/2),\ 0.990 + 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72521 - 0.192636i\)
\(L(\frac12)\) \(\approx\) \(2.72521 - 0.192636i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.89 + 1.19i)T \)
19 \( 1 + (3.50 + 2.58i)T \)
good7 \( 1 + (-2.32 - 2.32i)T + 7iT^{2} \)
11 \( 1 - 3.35T + 11T^{2} \)
13 \( 1 + (5.53 - 1.48i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.215 - 0.802i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-0.220 - 0.822i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.58 - 4.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.68iT - 31T^{2} \)
37 \( 1 + (-4.63 + 4.63i)T - 37iT^{2} \)
41 \( 1 + (-2.08 + 1.20i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.14 - 0.306i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (8.20 - 2.19i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.607 + 0.162i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.16 + 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.60 - 9.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.751 + 2.80i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.35 - 3.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.79 - 2.35i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.524 - 0.908i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.97 - 9.97i)T - 83iT^{2} \)
89 \( 1 + (-2.07 + 3.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.00 - 2.41i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.03324564756340393380094888267, −9.563295917223607191353793475204, −8.984630108374779381773851185697, −8.001098604423759037991175038175, −6.95306157716947613981351927542, −6.06463525041603650713297687643, −5.17709639515302907396775563888, −4.34149274750348793880001016457, −2.51848480616621042872495833701, −1.75050080552415123258467074722, 1.73306316411404861863142892762, 2.97367050657846881571611308900, 4.23676172198597878472449472912, 4.94125755912419396760536069786, 6.08032768087058667213602205637, 7.03193368753539199087963487729, 7.963879429931455554556421935793, 9.322397727863317446191041579313, 10.08043658943641501264174496172, 10.69512218042926083368502866304

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