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Properties

Label 2-570-15.2-c1-0-15
Degree $2$
Conductor $570$
Sign $0.599 + 0.800i$
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  + (−0.707 + 0.707i)2-s + (0.492 + 1.66i)3-s − 1.00i·4-s + (−1.86 + 1.23i)5-s + (−1.52 − 0.825i)6-s + (−3.33 − 3.33i)7-s + (0.707 + 0.707i)8-s + (−2.51 + 1.63i)9-s + (0.442 − 2.19i)10-s − 0.00329i·11-s + (1.66 − 0.492i)12-s + (1.14 − 1.14i)13-s + 4.71·14-s + (−2.97 − 2.48i)15-s − 1.00·16-s + (4.95 − 4.95i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.284 + 0.958i)3-s − 0.500i·4-s + (−0.833 + 0.553i)5-s + (−0.621 − 0.337i)6-s + (−1.26 − 1.26i)7-s + (0.250 + 0.250i)8-s + (−0.838 + 0.545i)9-s + (0.139 − 0.693i)10-s − 0.000994i·11-s + (0.479 − 0.142i)12-s + (0.318 − 0.318i)13-s + 1.26·14-s + (−0.767 − 0.641i)15-s − 0.250·16-s + (1.20 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.599 + 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391165 - 0.195736i\)
\(L(\frac12)\) \(\approx\) \(0.391165 - 0.195736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.492 - 1.66i)T \)
5 \( 1 + (1.86 - 1.23i)T \)
19 \( 1 + iT \)
good7 \( 1 + (3.33 + 3.33i)T + 7iT^{2} \)
11 \( 1 + 0.00329iT - 11T^{2} \)
13 \( 1 + (-1.14 + 1.14i)T - 13iT^{2} \)
17 \( 1 + (-4.95 + 4.95i)T - 17iT^{2} \)
23 \( 1 + (-1.28 - 1.28i)T + 23iT^{2} \)
29 \( 1 + 1.91T + 29T^{2} \)
31 \( 1 + 3.49T + 31T^{2} \)
37 \( 1 + (6.71 + 6.71i)T + 37iT^{2} \)
41 \( 1 - 3.63iT - 41T^{2} \)
43 \( 1 + (-1.52 + 1.52i)T - 43iT^{2} \)
47 \( 1 + (-2.07 + 2.07i)T - 47iT^{2} \)
53 \( 1 + (5.97 + 5.97i)T + 53iT^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 + 1.95T + 61T^{2} \)
67 \( 1 + (5.56 + 5.56i)T + 67iT^{2} \)
71 \( 1 - 3.22iT - 71T^{2} \)
73 \( 1 + (-10.3 + 10.3i)T - 73iT^{2} \)
79 \( 1 + 12.5iT - 79T^{2} \)
83 \( 1 + (9.29 + 9.29i)T + 83iT^{2} \)
89 \( 1 - 6.59T + 89T^{2} \)
97 \( 1 + (-7.72 - 7.72i)T + 97iT^{2} \)
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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

−10.51340973669441324201000447698, −9.710942255467691897570785758591, −9.042301305815838293047012920256, −7.71786736900538909043261281255, −7.31137035345087731010434611804, −6.20652390263172222207608771572, −4.94963585617650281404746434523, −3.69821897023410485632839176517, −3.14652535657717803062205872502, −0.29079062469094412409162009891, 1.48937791435181065499169942722, 2.93471809963806238344757753623, 3.72656603997696893713913244866, 5.55703468171631112268871803288, 6.44628442173187014273945436376, 7.52247689664031116907710841500, 8.422634609429621876170281676870, 8.923351277397706979924440004293, 9.772796638311513704312913927619, 11.02800387057434960439149494778

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