Properties

Label 2-572-572.571-c1-0-37
Degree $2$
Conductor $572$
Sign $0.639 - 0.768i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
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  + 1.41i·2-s − 2.00·4-s − 1.41i·7-s − 2.82i·8-s + 3·9-s + (2.54 + 2.12i)11-s − 3.60i·13-s + 2.00·14-s + 4.00·16-s − 7.21i·17-s + 4.24i·18-s + 7.07i·19-s + (−3 + 3.60i)22-s + 5·25-s + 5.09·26-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s − 0.534i·7-s − 1.00i·8-s + 9-s + (0.768 + 0.639i)11-s − 0.999i·13-s + 0.534·14-s + 1.00·16-s − 1.74i·17-s + 0.999i·18-s + 1.62i·19-s + (−0.639 + 0.768i)22-s + 25-s + 0.999·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.639 - 0.768i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.639 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32400 + 0.620743i\)
\(L(\frac12)\) \(\approx\) \(1.32400 + 0.620743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
11 \( 1 + (-2.54 - 2.12i)T \)
13 \( 1 + 3.60iT \)
good3 \( 1 - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 1.41iT - 7T^{2} \)
17 \( 1 + 7.21iT - 17T^{2} \)
19 \( 1 - 7.07iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 7.21iT - 29T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 5.09T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 15.2T + 59T^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 + 5.09T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 9.89iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.44090836647967685651339972315, −9.925345488281828090567661347160, −9.062013932034061457061163171055, −7.942217907173544706988870438423, −7.20631296219976137619443419523, −6.58815029907061836577563607635, −5.27826290059590094971809743771, −4.47566090405179387501475894352, −3.39559670026226394060869617387, −1.12871449318928671388025298562, 1.27546066207092635188120514625, 2.52981946416455970061037688243, 3.93269919705747054175798928512, 4.58863696712385945258153389738, 5.97467967240815976958241322479, 6.94116333386217245079951902371, 8.400048867417349229851167183087, 8.950426771650955963201684786207, 9.816856320976323767423082738147, 10.67613271856637668358488398406

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