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Properties

Label 2-1470-35.13-c1-0-24
Degree $2$
Conductor $1470$
Sign $0.652 + 0.757i$
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  + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (1 − 2i)5-s − 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (0.707 + 2.12i)10-s + 4.79·11-s + (0.707 + 0.707i)12-s + (4.37 − 4.37i)13-s + (0.707 + 2.12i)15-s − 1.00·16-s + (−1.38 − 1.38i)17-s + (0.707 + 0.707i)18-s − 3.41·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.447 − 0.894i)5-s − 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.223 + 0.670i)10-s + 1.44·11-s + (0.204 + 0.204i)12-s + (1.21 − 1.21i)13-s + (0.182 + 0.547i)15-s − 0.250·16-s + (−0.336 − 0.336i)17-s + (0.166 + 0.166i)18-s − 0.783·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.652 + 0.757i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.652 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.177613070\)
\(L(\frac12)\) \(\approx\) \(1.177613070\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1 + 2i)T \)
7 \( 1 \)
good11 \( 1 - 4.79T + 11T^{2} \)
13 \( 1 + (-4.37 + 4.37i)T - 13iT^{2} \)
17 \( 1 + (1.38 + 1.38i)T + 17iT^{2} \)
19 \( 1 + 3.41T + 19T^{2} \)
23 \( 1 + (3 + 3i)T + 23iT^{2} \)
29 \( 1 - 1.42iT - 29T^{2} \)
31 \( 1 + 0.813iT - 31T^{2} \)
37 \( 1 + (6.18 - 6.18i)T - 37iT^{2} \)
41 \( 1 + 1.92iT - 41T^{2} \)
43 \( 1 + (-5.83 - 5.83i)T + 43iT^{2} \)
47 \( 1 + (6.59 + 6.59i)T + 47iT^{2} \)
53 \( 1 + (7.91 + 7.91i)T + 53iT^{2} \)
59 \( 1 - 5.36T + 59T^{2} \)
61 \( 1 + 9.58iT - 61T^{2} \)
67 \( 1 + (6.36 - 6.36i)T - 67iT^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + (-4.81 + 4.81i)T - 73iT^{2} \)
79 \( 1 - 1.92iT - 79T^{2} \)
83 \( 1 + (-2.51 + 2.51i)T - 83iT^{2} \)
89 \( 1 - 9.97T + 89T^{2} \)
97 \( 1 + (11.2 + 11.2i)T + 97iT^{2} \)
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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

−9.339371475214201362297905751385, −8.554361018113314091899433039315, −8.162067491531444797620495484734, −6.65545406141931771369635367634, −6.24163924666087349552161248384, −5.36677940278197182716603585874, −4.50861622752727608426762122222, −3.54701245534554717381348329346, −1.74943830611999937697848543888, −0.63121370802193599690620180201, 1.42966963180544651504645276670, 2.12770463787881845381308387407, 3.59399857186937954779404937248, 4.22239715533603884620287252671, 5.86784652069131523675778118622, 6.49429797715981883053176537469, 6.99487252681793728733615607491, 8.095400240842854334908087969185, 9.064864894015106943004074231107, 9.480652528775249151790330703017

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