Properties

Label 2-570-95.18-c1-0-13
Degree $2$
Conductor $570$
Sign $0.781 - 0.624i$
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 + (0.707 − 0.707i)3-s + 1.00i·4-s + (1.89 + 1.18i)5-s + 1.00·6-s + (0.705 − 0.705i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (0.498 + 2.17i)10-s + 1.32·11-s + (0.707 + 0.707i)12-s + (−0.741 + 0.741i)13-s + 0.997·14-s + (2.17 − 0.498i)15-s − 1.00·16-s + (2.17 − 2.17i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + (0.846 + 0.531i)5-s + 0.408·6-s + (0.266 − 0.266i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.157 + 0.689i)10-s + 0.400·11-s + (0.204 + 0.204i)12-s + (−0.205 + 0.205i)13-s + 0.266·14-s + (0.562 − 0.128i)15-s − 0.250·16-s + (0.527 − 0.527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.781 - 0.624i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.781 - 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37296 + 0.831412i\)
\(L(\frac12)\) \(\approx\) \(2.37296 + 0.831412i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-1.89 - 1.18i)T \)
19 \( 1 + (-0.939 - 4.25i)T \)
good7 \( 1 + (-0.705 + 0.705i)T - 7iT^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 + (0.741 - 0.741i)T - 13iT^{2} \)
17 \( 1 + (-2.17 + 2.17i)T - 17iT^{2} \)
23 \( 1 + (1.08 + 1.08i)T + 23iT^{2} \)
29 \( 1 + 5.95T + 29T^{2} \)
31 \( 1 + 5.95iT - 31T^{2} \)
37 \( 1 + (1.33 + 1.33i)T + 37iT^{2} \)
41 \( 1 - 0.531iT - 41T^{2} \)
43 \( 1 + (-1.53 - 1.53i)T + 43iT^{2} \)
47 \( 1 + (-4.13 + 4.13i)T - 47iT^{2} \)
53 \( 1 + (5.48 - 5.48i)T - 53iT^{2} \)
59 \( 1 + 3.53T + 59T^{2} \)
61 \( 1 - 3.41T + 61T^{2} \)
67 \( 1 + (1.99 + 1.99i)T + 67iT^{2} \)
71 \( 1 + 8.71iT - 71T^{2} \)
73 \( 1 + (5.83 + 5.83i)T + 73iT^{2} \)
79 \( 1 - 9.31T + 79T^{2} \)
83 \( 1 + (9.50 + 9.50i)T + 83iT^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + (-1.11 - 1.11i)T + 97iT^{2} \)
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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

−10.87049411847062266135677763902, −9.792703265354200993903734147411, −9.102963083569047185883538803953, −7.84957472071421494484495204010, −7.26343283644392345290857795945, −6.24829570272912504611425501847, −5.52981481156469762288793343817, −4.17048860106896655068903385197, −3.02951565637652498796341287268, −1.76703155740687037480109764950, 1.51554366828672351667668815223, 2.70380568544062876593248444476, 3.93234267097739576873019591835, 5.04495103223668505702418685952, 5.68006057130780219968182550149, 6.90863661534857191240450600815, 8.265262894391249107127483005711, 9.114113402766078714002406521622, 9.764251154418862641154543405727, 10.59543946260803188063944123968

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