Properties

Label 2-570-15.8-c1-0-31
Degree $2$
Conductor $570$
Sign $0.865 - 0.501i$
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 + (1.72 − 0.0916i)3-s + 1.00i·4-s + (2.17 − 0.532i)5-s + (1.28 + 1.15i)6-s + (1.32 − 1.32i)7-s + (−0.707 + 0.707i)8-s + (2.98 − 0.316i)9-s + (1.91 + 1.15i)10-s + 4.14i·11-s + (0.0916 + 1.72i)12-s + (−3.92 − 3.92i)13-s + 1.87·14-s + (3.70 − 1.11i)15-s − 1.00·16-s + (−5.01 − 5.01i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.998 − 0.0529i)3-s + 0.500i·4-s + (0.971 − 0.237i)5-s + (0.525 + 0.472i)6-s + (0.500 − 0.500i)7-s + (−0.250 + 0.250i)8-s + (0.994 − 0.105i)9-s + (0.604 + 0.366i)10-s + 1.25i·11-s + (0.0264 + 0.499i)12-s + (−1.08 − 1.08i)13-s + 0.500·14-s + (0.957 − 0.289i)15-s − 0.250·16-s + (−1.21 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.865 - 0.501i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.865 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86954 + 0.771126i\)
\(L(\frac12)\) \(\approx\) \(2.86954 + 0.771126i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.0916i)T \)
5 \( 1 + (-2.17 + 0.532i)T \)
19 \( 1 - iT \)
good7 \( 1 + (-1.32 + 1.32i)T - 7iT^{2} \)
11 \( 1 - 4.14iT - 11T^{2} \)
13 \( 1 + (3.92 + 3.92i)T + 13iT^{2} \)
17 \( 1 + (5.01 + 5.01i)T + 17iT^{2} \)
23 \( 1 + (5.46 - 5.46i)T - 23iT^{2} \)
29 \( 1 + 0.0964T + 29T^{2} \)
31 \( 1 - 0.0111T + 31T^{2} \)
37 \( 1 + (-1.55 + 1.55i)T - 37iT^{2} \)
41 \( 1 - 8.45iT - 41T^{2} \)
43 \( 1 + (-0.130 - 0.130i)T + 43iT^{2} \)
47 \( 1 + (-1.71 - 1.71i)T + 47iT^{2} \)
53 \( 1 + (8.88 - 8.88i)T - 53iT^{2} \)
59 \( 1 - 6.40T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + (-4.77 + 4.77i)T - 67iT^{2} \)
71 \( 1 + 9.47iT - 71T^{2} \)
73 \( 1 + (4.04 + 4.04i)T + 73iT^{2} \)
79 \( 1 + 12.4iT - 79T^{2} \)
83 \( 1 + (1.43 - 1.43i)T - 83iT^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + (-3.48 + 3.48i)T - 97iT^{2} \)
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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

−10.55957558800915878028479203662, −9.679260118868661271531110273318, −9.192880373761102918478587394938, −7.75672712854085389319979864364, −7.50570309458092473194524978178, −6.34334124034445035155350000871, −4.98336087145576979549339073584, −4.45125834670236603596754501423, −2.87824355739887557248224842844, −1.88204295425163607591207412895, 1.94782074739240037101810103405, 2.46620448552354939497710319323, 3.86627374842928826284325829005, 4.86536940845650109083549313969, 6.07110759583453032963074839486, 6.88068899624336005396642025237, 8.393304663235531660376695649374, 8.889668404944585686878895419128, 9.839903609600930457177338419683, 10.60382229349736504717695852528

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