Properties

Label 2-570-95.64-c1-0-15
Degree $2$
Conductor $570$
Sign $-0.435 + 0.900i$
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  + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−1.34 + 1.78i)5-s + (−0.499 + 0.866i)6-s − 4.03i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.271 − 2.21i)10-s − 1.47·11-s − 0.999i·12-s + (−4.38 − 2.53i)13-s + (2.01 + 3.49i)14-s + (−0.271 + 2.21i)15-s + (−0.5 − 0.866i)16-s + (−0.0812 + 0.0469i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.601 + 0.798i)5-s + (−0.204 + 0.353i)6-s − 1.52i·7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.0859 − 0.701i)10-s − 0.443·11-s − 0.288i·12-s + (−1.21 − 0.701i)13-s + (0.539 + 0.933i)14-s + (−0.0701 + 0.573i)15-s + (−0.125 − 0.216i)16-s + (−0.0197 + 0.0113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.435 + 0.900i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.435 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314231 - 0.501097i\)
\(L(\frac12)\) \(\approx\) \(0.314231 - 0.501097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (1.34 - 1.78i)T \)
19 \( 1 + (4.00 - 1.72i)T \)
good7 \( 1 + 4.03iT - 7T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 + (4.38 + 2.53i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.0812 - 0.0469i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.22 + 1.28i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.10 + 3.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.26T + 31T^{2} \)
37 \( 1 + 1.53iT - 37T^{2} \)
41 \( 1 + (3.88 + 6.72i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.97 - 2.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.49 - 2.01i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.22 + 5.32i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.07 + 5.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.653 + 1.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.31 - 5.37i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.33 - 7.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-11.0 + 6.35i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.48 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.06iT - 83T^{2} \)
89 \( 1 + (0.813 - 1.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.3 + 5.97i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.27765467682152277182716580393, −9.848292221501339973541088900593, −8.277941097068598786986087482417, −7.80333076424960375743715994385, −7.10915105567034353799436291055, −6.34983060943065890119412852987, −4.70404714270378827474236292392, −3.59532228827846163443272033662, −2.33807003843672439927529322718, −0.36213723560067244779896110422, 1.98323027957190965900444698109, 2.95588211002608525063142582412, 4.43061960899959508433049649444, 5.24712476443241563713339547132, 6.68766454385059680316361987815, 7.927687694236427301598786046152, 8.496762761630007820686668815634, 9.223866774469694988515509660290, 9.839515630475239072838862774875, 11.03599505683573303945304194270

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