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Properties

Label 2-570-95.37-c1-0-13
Degree $2$
Conductor $570$
Sign $-0.978 + 0.204i$
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 + (−0.253 − 2.22i)5-s + 1.00·6-s + (−2.47 − 2.47i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (1.75 + 1.39i)10-s + 2.74·11-s + (−0.707 + 0.707i)12-s + (1.20 + 1.20i)13-s + 3.50·14-s + (−1.39 + 1.75i)15-s − 1.00·16-s + (−4.87 − 4.87i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (−0.113 − 0.993i)5-s + 0.408·6-s + (−0.935 − 0.935i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.553 + 0.440i)10-s + 0.827·11-s + (−0.204 + 0.204i)12-s + (0.333 + 0.333i)13-s + 0.935·14-s + (−0.359 + 0.451i)15-s − 0.250·16-s + (−1.18 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.978 + 0.204i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.978 + 0.204i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0347170 - 0.335798i\)
\(L(\frac12)\) \(\approx\) \(0.0347170 - 0.335798i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.253 + 2.22i)T \)
19 \( 1 + (1.94 - 3.90i)T \)
good7 \( 1 + (2.47 + 2.47i)T + 7iT^{2} \)
11 \( 1 - 2.74T + 11T^{2} \)
13 \( 1 + (-1.20 - 1.20i)T + 13iT^{2} \)
17 \( 1 + (4.87 + 4.87i)T + 17iT^{2} \)
23 \( 1 + (-0.0321 + 0.0321i)T - 23iT^{2} \)
29 \( 1 + 6.50T + 29T^{2} \)
31 \( 1 - 6.50iT - 31T^{2} \)
37 \( 1 + (4.58 - 4.58i)T - 37iT^{2} \)
41 \( 1 + 5.96iT - 41T^{2} \)
43 \( 1 + (-5.39 + 5.39i)T - 43iT^{2} \)
47 \( 1 + (-3.66 - 3.66i)T + 47iT^{2} \)
53 \( 1 + (8.97 + 8.97i)T + 53iT^{2} \)
59 \( 1 + 4.42T + 59T^{2} \)
61 \( 1 + 2.95T + 61T^{2} \)
67 \( 1 + (7.00 - 7.00i)T - 67iT^{2} \)
71 \( 1 + 5.56iT - 71T^{2} \)
73 \( 1 + (2.19 - 2.19i)T - 73iT^{2} \)
79 \( 1 - 0.225T + 79T^{2} \)
83 \( 1 + (3.87 - 3.87i)T - 83iT^{2} \)
89 \( 1 - 9.13T + 89T^{2} \)
97 \( 1 + (-8.76 + 8.76i)T - 97iT^{2} \)
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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

−10.20194332283918183695827840716, −9.244916884726688845082416293918, −8.711946488219374594093827103545, −7.45893944108351912983940097941, −6.81116567051728899799482403202, −5.98240595059851871494970012418, −4.77860042346309242565187169350, −3.76378153345659601048576948569, −1.60803696079533543218176037815, −0.23468687304048084947566864387, 2.20155277323086302692439126398, 3.32465009904882975528501660196, 4.25863970468387340597851830335, 6.05272938760906316874465102481, 6.40149704841424157770650464755, 7.61764982393746041993674641686, 8.982238663800216063094836818911, 9.306122883350531058024011912942, 10.43388350257403678961707544781, 11.03764490537274373075150873412

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