Properties

Label 2-1470-7.6-c2-0-51
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 − 0.526·11-s − 3.46i·12-s + 4.22i·13-s − 3.87·15-s + 4.00·16-s − 33.3i·17-s − 4.24·18-s − 3.17i·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 − 0.0478·11-s − 0.288i·12-s + 0.324i·13-s − 0.258·15-s + 0.250·16-s − 1.96i·17-s − 0.235·18-s − 0.167i·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

\(L(\frac{3}{2})\) \(\approx\) \(2.234548778\)
\(L(\frac12)\) \(\approx\) \(2.234548778\)
\(L(2)\) 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 \( 1 + 0.526T + 121T^{2} \)
13 \( 1 - 4.22iT - 169T^{2} \)
17 \( 1 + 33.3iT - 289T^{2} \)
19 \( 1 + 3.17iT - 361T^{2} \)
23 \( 1 - 11.0T + 529T^{2} \)
29 \( 1 + 56.1T + 841T^{2} \)
31 \( 1 + 1.89iT - 961T^{2} \)
37 \( 1 + 9.75T + 1.36e3T^{2} \)
41 \( 1 + 4.07iT - 1.68e3T^{2} \)
43 \( 1 - 46.3T + 1.84e3T^{2} \)
47 \( 1 + 63.2iT - 2.20e3T^{2} \)
53 \( 1 + 46.5T + 2.80e3T^{2} \)
59 \( 1 + 50.5iT - 3.48e3T^{2} \)
61 \( 1 + 103. iT - 3.72e3T^{2} \)
67 \( 1 - 8.73T + 4.48e3T^{2} \)
71 \( 1 - 29.0T + 5.04e3T^{2} \)
73 \( 1 - 16.7iT - 5.32e3T^{2} \)
79 \( 1 + 132.T + 6.24e3T^{2} \)
83 \( 1 + 12.4iT - 6.88e3T^{2} \)
89 \( 1 - 68.2iT - 7.92e3T^{2} \)
97 \( 1 - 149. iT - 9.40e3T^{2} \)
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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

−9.120761801834733647109422653098, −7.986475870531115452520212818998, −7.25247460095410181263273853241, −6.63441912487355708686290712843, −5.49353923776338410214948676033, −4.99702579835530139686165816651, −3.87854845661579663190037706377, −2.81577357451300330301509742326, −1.80966884588449242355823741464, −0.45284917150594097425899327994, 1.62872122659042418256812188272, 2.85542412113483536002625149849, 3.75335674239168781416814204910, 4.40724000140471499279769979100, 5.65341951903731152707880340949, 6.01378682543008231342205045737, 7.14989090813468079167393497877, 7.933368934919688949831017266200, 8.851333574853433004299466588662, 9.768403726681270071331650912295

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