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Properties

Label 2-1470-7.6-c2-0-0
Degree $2$
Conductor $1470$
Sign $-0.755 + 0.654i$
Analytic cond. $40.0545$
Root an. cond. $6.32887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 1.73i·3-s + 2.00·4-s + 2.23i·5-s + 2.44i·6-s + 2.82·8-s − 2.99·9-s + 3.16i·10-s − 18.0·11-s + 3.46i·12-s + 18.6i·13-s − 3.87·15-s + 4.00·16-s + 1.54i·17-s − 4.24·18-s − 34.0i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.447i·5-s + 0.408i·6-s + 0.353·8-s − 0.333·9-s + 0.316i·10-s − 1.64·11-s + 0.288i·12-s + 1.43i·13-s − 0.258·15-s + 0.250·16-s + 0.0906i·17-s − 0.235·18-s − 1.79i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.755 + 0.654i$
Analytic conductor: \(40.0545\)
Root analytic conductor: \(6.32887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1),\ -0.755 + 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1754075425\)
\(L(\frac12)\) \(\approx\) \(0.1754075425\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 - 1.73iT \)
5 \( 1 - 2.23iT \)
7 \( 1 \)
good11 \( 1 + 18.0T + 121T^{2} \)
13 \( 1 - 18.6iT - 169T^{2} \)
17 \( 1 - 1.54iT - 289T^{2} \)
19 \( 1 + 34.0iT - 361T^{2} \)
23 \( 1 + 26.9T + 529T^{2} \)
29 \( 1 + 16.4T + 841T^{2} \)
31 \( 1 + 27.9iT - 961T^{2} \)
37 \( 1 - 51.6T + 1.36e3T^{2} \)
41 \( 1 + 37.4iT - 1.68e3T^{2} \)
43 \( 1 + 63.6T + 1.84e3T^{2} \)
47 \( 1 - 28.2iT - 2.20e3T^{2} \)
53 \( 1 - 0.443T + 2.80e3T^{2} \)
59 \( 1 - 74.1iT - 3.48e3T^{2} \)
61 \( 1 + 105. iT - 3.72e3T^{2} \)
67 \( 1 + 12.7T + 4.48e3T^{2} \)
71 \( 1 + 45.7T + 5.04e3T^{2} \)
73 \( 1 - 36.4iT - 5.32e3T^{2} \)
79 \( 1 + 133.T + 6.24e3T^{2} \)
83 \( 1 - 49.9iT - 6.88e3T^{2} \)
89 \( 1 + 99.2iT - 7.92e3T^{2} \)
97 \( 1 - 150. iT - 9.40e3T^{2} \)
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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

−9.893726702973051125827823489768, −9.156189605319743216822336865956, −8.084353947373999914629836774660, −7.29890670568250570636426893572, −6.44262083049373051036179701076, −5.56598081283311027521892461169, −4.70471540100339654538997451560, −4.02686630015366453741981375668, −2.83787978858467602744694365905, −2.15260195735725743400243430695, 0.03389778743437913129245694015, 1.53301702801071070697704962607, 2.66641618426686005097644732907, 3.53068423537771859888445957655, 4.76702371160003989469181020525, 5.63084937543305616308474514401, 5.99402142414874229728799136575, 7.34205541133454697247270408347, 8.016339256100757578390958646714, 8.354329071793163272699628273288

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