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Properties

Label 2-1530-5.4-c1-0-11
Degree $2$
Conductor $1530$
Sign $0.970 + 0.241i$
Analytic cond. $12.2171$
Root an. cond. $3.49529$
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 − 4-s + (−2.17 − 0.539i)5-s − 1.07i·7-s + i·8-s + (−0.539 + 2.17i)10-s + 4·11-s + 5.41i·13-s − 1.07·14-s + 16-s − i·17-s − 8.68·19-s + (2.17 + 0.539i)20-s − 4i·22-s + 0.921i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.970 − 0.241i)5-s − 0.407i·7-s + 0.353i·8-s + (−0.170 + 0.686i)10-s + 1.20·11-s + 1.50i·13-s − 0.288·14-s + 0.250·16-s − 0.242i·17-s − 1.99·19-s + (0.485 + 0.120i)20-s − 0.852i·22-s + 0.192i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.970 + 0.241i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.202790864\)
\(L(\frac12)\) \(\approx\) \(1.202790864\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.17 + 0.539i)T \)
17 \( 1 + iT \)
good7 \( 1 + 1.07iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 5.41iT - 13T^{2} \)
19 \( 1 + 8.68T + 19T^{2} \)
23 \( 1 - 0.921iT - 23T^{2} \)
29 \( 1 - 4.34T + 29T^{2} \)
31 \( 1 - 3.07T + 31T^{2} \)
37 \( 1 - 8.34iT - 37T^{2} \)
41 \( 1 - 3.26T + 41T^{2} \)
43 \( 1 - 9.41iT - 43T^{2} \)
47 \( 1 + 2.15iT - 47T^{2} \)
53 \( 1 + 13.5iT - 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 7.23T + 61T^{2} \)
67 \( 1 - 2.58iT - 67T^{2} \)
71 \( 1 - 3.60T + 71T^{2} \)
73 \( 1 + 6.58iT - 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 8.68iT - 83T^{2} \)
89 \( 1 - 6.15T + 89T^{2} \)
97 \( 1 - 2.09iT - 97T^{2} \)
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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

−9.414744734026101829189217356741, −8.668727354208290471065811233927, −8.126533496391975444265344838231, −6.82625080765641625841973916194, −6.48233341241629766129742070061, −4.78856400403204038713692570245, −4.23535049240230229610147720438, −3.62025878215266395683639910820, −2.20865440168209913951849714082, −0.992457550403183386729955728712, 0.62756135226311544364931957310, 2.52565330898502292967516854365, 3.77911569648131940269469090822, 4.34082753961749190299647259119, 5.52724213919738405375905769996, 6.34356887320025891768554116049, 7.02990092557533848016824623314, 7.974921935509109493391655900837, 8.531515600231598253713721216823, 9.152358907486455803077493215189

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