Properties

Label 2-570-15.2-c1-0-13
Degree $2$
Conductor $570$
Sign $0.775 - 0.631i$
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 + (0.867 + 1.49i)3-s − 1.00i·4-s + (0.850 + 2.06i)5-s + (1.67 + 0.446i)6-s + (0.811 + 0.811i)7-s + (−0.707 − 0.707i)8-s + (−1.49 + 2.60i)9-s + (2.06 + 0.861i)10-s − 1.25i·11-s + (1.49 − 0.867i)12-s + (3.40 − 3.40i)13-s + 1.14·14-s + (−2.36 + 3.06i)15-s − 1.00·16-s + (−3.04 + 3.04i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.500 + 0.865i)3-s − 0.500i·4-s + (0.380 + 0.924i)5-s + (0.683 + 0.182i)6-s + (0.306 + 0.306i)7-s + (−0.250 − 0.250i)8-s + (−0.498 + 0.867i)9-s + (0.652 + 0.272i)10-s − 0.377i·11-s + (0.432 − 0.250i)12-s + (0.944 − 0.944i)13-s + 0.306·14-s + (−0.610 + 0.792i)15-s − 0.250·16-s + (−0.737 + 0.737i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.775 - 0.631i$
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: \((2,\ 570,\ (\ :1/2),\ 0.775 - 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23940 + 0.795803i\)
\(L(\frac12)\) \(\approx\) \(2.23940 + 0.795803i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.867 - 1.49i)T \)
5 \( 1 + (-0.850 - 2.06i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-0.811 - 0.811i)T + 7iT^{2} \)
11 \( 1 + 1.25iT - 11T^{2} \)
13 \( 1 + (-3.40 + 3.40i)T - 13iT^{2} \)
17 \( 1 + (3.04 - 3.04i)T - 17iT^{2} \)
23 \( 1 + (-5.51 - 5.51i)T + 23iT^{2} \)
29 \( 1 - 5.49T + 29T^{2} \)
31 \( 1 + 5.80T + 31T^{2} \)
37 \( 1 + (7.08 + 7.08i)T + 37iT^{2} \)
41 \( 1 - 0.259iT - 41T^{2} \)
43 \( 1 + (-5.60 + 5.60i)T - 43iT^{2} \)
47 \( 1 + (-6.12 + 6.12i)T - 47iT^{2} \)
53 \( 1 + (0.465 + 0.465i)T + 53iT^{2} \)
59 \( 1 + 9.08T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 + (2.40 + 2.40i)T + 67iT^{2} \)
71 \( 1 - 0.747iT - 71T^{2} \)
73 \( 1 + (-5.45 + 5.45i)T - 73iT^{2} \)
79 \( 1 - 4.84iT - 79T^{2} \)
83 \( 1 + (4.22 + 4.22i)T + 83iT^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (12.5 + 12.5i)T + 97iT^{2} \)
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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

−10.82608619097721990582132431870, −10.27656020046323854732114042219, −9.156353274242218182182889278439, −8.475764070074400662945877567347, −7.22224112380885562474202716933, −5.89465704343369640021846760534, −5.29131839239355801780258972870, −3.82769275145354118365314645107, −3.19901920914129738681001608528, −2.01290735436846636147843742689, 1.26617833647278509302043991611, 2.63204954270154254421751256607, 4.17330457316052785556349225736, 5.00025016655674006944153648587, 6.30667942306118108792359020920, 6.90303697786915612312697854360, 7.936877319512891787286359942112, 8.870343476006355280950216626016, 9.213794685713038139365398418861, 10.83313439806688216912644449852

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