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Properties

Label 2-570-15.2-c1-0-32
Degree $2$
Conductor $570$
Sign $-0.947 + 0.320i$
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 + (1.52 − 0.830i)3-s − 1.00i·4-s + (−2.23 + 0.129i)5-s + (−0.487 + 1.66i)6-s + (−2.47 − 2.47i)7-s + (0.707 + 0.707i)8-s + (1.62 − 2.52i)9-s + (1.48 − 1.66i)10-s − 0.319i·11-s + (−0.830 − 1.52i)12-s + (−3.95 + 3.95i)13-s + 3.49·14-s + (−3.28 + 2.04i)15-s − 1.00·16-s + (−4.30 + 4.30i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.877 − 0.479i)3-s − 0.500i·4-s + (−0.998 + 0.0577i)5-s + (−0.199 + 0.678i)6-s + (−0.935 − 0.935i)7-s + (0.250 + 0.250i)8-s + (0.540 − 0.841i)9-s + (0.470 − 0.528i)10-s − 0.0963i·11-s + (−0.239 − 0.438i)12-s + (−1.09 + 1.09i)13-s + 0.935·14-s + (−0.848 + 0.529i)15-s − 0.250·16-s + (−1.04 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.947 + 0.320i$
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: \((2,\ 570,\ (\ :1/2),\ -0.947 + 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0370401 - 0.225253i\)
\(L(\frac12)\) \(\approx\) \(0.0370401 - 0.225253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 0.830i)T \)
5 \( 1 + (2.23 - 0.129i)T \)
19 \( 1 - iT \)
good7 \( 1 + (2.47 + 2.47i)T + 7iT^{2} \)
11 \( 1 + 0.319iT - 11T^{2} \)
13 \( 1 + (3.95 - 3.95i)T - 13iT^{2} \)
17 \( 1 + (4.30 - 4.30i)T - 17iT^{2} \)
23 \( 1 + (2.89 + 2.89i)T + 23iT^{2} \)
29 \( 1 + 8.60T + 29T^{2} \)
31 \( 1 + 2.73T + 31T^{2} \)
37 \( 1 + (-1.42 - 1.42i)T + 37iT^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + (-6.45 + 6.45i)T - 43iT^{2} \)
47 \( 1 + (4.12 - 4.12i)T - 47iT^{2} \)
53 \( 1 + (-1.15 - 1.15i)T + 53iT^{2} \)
59 \( 1 - 7.51T + 59T^{2} \)
61 \( 1 - 8.09T + 61T^{2} \)
67 \( 1 + (11.1 + 11.1i)T + 67iT^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (9.22 - 9.22i)T - 73iT^{2} \)
79 \( 1 + 3.00iT - 79T^{2} \)
83 \( 1 + (-2.31 - 2.31i)T + 83iT^{2} \)
89 \( 1 + 4.95T + 89T^{2} \)
97 \( 1 + (2.21 + 2.21i)T + 97iT^{2} \)
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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

−10.13513606456822277916224753703, −9.250639741981146374125650928843, −8.548760812419329372254457398947, −7.46504636575831595366103180004, −7.11807005110899735417754475476, −6.25108866726231655003577178922, −4.33400277040361922650208985289, −3.68195792917966577254115990226, −2.08876438103708554598132715276, −0.12733566511542629729485081357, 2.45518227955861953644485432152, 3.12989840549974088232513594389, 4.23428597025294835104932619967, 5.39749024914230510037343817795, 7.10331204843888866796800522089, 7.74488796283724494256954862535, 8.698215479190079770988074262586, 9.416184312020898818105065850535, 9.963183374604174143128901914116, 11.12048665574468505107951076675

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