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Properties

Label 2-1470-21.20-c1-0-46
Degree $2$
Conductor $1470$
Sign $0.872 + 0.488i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
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  + i·2-s + (−0.152 + 1.72i)3-s − 4-s + 5-s + (−1.72 − 0.152i)6-s − i·8-s + (−2.95 − 0.527i)9-s + i·10-s − 0.476i·11-s + (0.152 − 1.72i)12-s − 4.96i·13-s + (−0.152 + 1.72i)15-s + 16-s − 5.31·17-s + (0.527 − 2.95i)18-s − 5.18i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.0882 + 0.996i)3-s − 0.5·4-s + 0.447·5-s + (−0.704 − 0.0623i)6-s − 0.353i·8-s + (−0.984 − 0.175i)9-s + 0.316i·10-s − 0.143i·11-s + (0.0441 − 0.498i)12-s − 1.37i·13-s + (−0.0394 + 0.445i)15-s + 0.250·16-s − 1.28·17-s + (0.124 − 0.696i)18-s − 1.18i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8385204698\)
\(L(\frac12)\) \(\approx\) \(0.8385204698\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.152 - 1.72i)T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + 0.476iT - 11T^{2} \)
13 \( 1 + 4.96iT - 13T^{2} \)
17 \( 1 + 5.31T + 17T^{2} \)
19 \( 1 + 5.18iT - 19T^{2} \)
23 \( 1 + 3.71iT - 23T^{2} \)
29 \( 1 + 4.80iT - 29T^{2} \)
31 \( 1 - 2.47iT - 31T^{2} \)
37 \( 1 - 2.38T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 1.38T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 - 0.305iT - 53T^{2} \)
59 \( 1 - 0.441T + 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 - 3.11T + 67T^{2} \)
71 \( 1 + 8.78iT - 71T^{2} \)
73 \( 1 - 15.0iT - 73T^{2} \)
79 \( 1 - 1.77T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + 5.22T + 89T^{2} \)
97 \( 1 - 8.50iT - 97T^{2} \)
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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

−9.419107620309683805189225507504, −8.605262598911051124905299029266, −8.080014784684310911695066282747, −6.76934009791021888743882992763, −6.17922085593267352286116291329, −5.13767744158795219407665251471, −4.71966222938714262874612688313, −3.52169342398316785192652368110, −2.51778912592420909112705068206, −0.32671909315549429437109997722, 1.54126743108257595236409530085, 2.07615244783762639408058327407, 3.31952323514257472707598494647, 4.45781652129014011318790455216, 5.46544060900647266605092094828, 6.42343817562299544717628065411, 7.02431676934624821546094324203, 8.088458031766913693017846343456, 8.850959245805087035427374565805, 9.516879253843987585785373967394

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