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Properties

Label 2-1470-105.104-c1-0-57
Degree $2$
Conductor $1470$
Sign $0.994 + 0.100i$
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  + 2-s + (1.71 + 0.240i)3-s + 4-s + (0.263 − 2.22i)5-s + (1.71 + 0.240i)6-s + 8-s + (2.88 + 0.826i)9-s + (0.263 − 2.22i)10-s + 4.81i·11-s + (1.71 + 0.240i)12-s + 4.56·13-s + (0.986 − 3.74i)15-s + 16-s − 1.16i·17-s + (2.88 + 0.826i)18-s + 4.61i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.990 + 0.139i)3-s + 0.5·4-s + (0.117 − 0.993i)5-s + (0.700 + 0.0983i)6-s + 0.353·8-s + (0.961 + 0.275i)9-s + (0.0832 − 0.702i)10-s + 1.45i·11-s + (0.495 + 0.0695i)12-s + 1.26·13-s + (0.254 − 0.967i)15-s + 0.250·16-s − 0.282i·17-s + (0.679 + 0.194i)18-s + 1.05i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.994 + 0.100i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.026486401\)
\(L(\frac12)\) \(\approx\) \(4.026486401\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.71 - 0.240i)T \)
5 \( 1 + (-0.263 + 2.22i)T \)
7 \( 1 \)
good11 \( 1 - 4.81iT - 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 + 1.16iT - 17T^{2} \)
19 \( 1 - 4.61iT - 19T^{2} \)
23 \( 1 + 3.85T + 23T^{2} \)
29 \( 1 + 1.96iT - 29T^{2} \)
31 \( 1 + 5.57iT - 31T^{2} \)
37 \( 1 + 8.98iT - 37T^{2} \)
41 \( 1 + 3.76T + 41T^{2} \)
43 \( 1 - 4.02iT - 43T^{2} \)
47 \( 1 + 6.36iT - 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 4.96iT - 61T^{2} \)
67 \( 1 - 12.5iT - 67T^{2} \)
71 \( 1 + 2.74iT - 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 3.54iT - 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + 1.30T + 97T^{2} \)
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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

−9.658867479133886769514026934261, −8.552213252627851209003039192790, −7.995380993795134928896119945514, −7.16386760606643182802401277413, −6.08405543278979004018649807728, −5.17627407218972254935534212301, −4.15280240525567434246797944437, −3.81337253994005939721838359198, −2.29397872179399618733529091739, −1.52191830326236710371607288891, 1.46595102027102474303648102205, 2.80200676564151895533458018448, 3.32449290611249120962931169320, 4.09942983759754234325007682332, 5.45643273595033824856574340467, 6.45914895641378377226901084384, 6.81112643227512530783815556031, 8.052476845664914272924188717870, 8.485638727509201122662303894990, 9.481263854564069004514777624222

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