Properties

Degree $2$
Conductor $1470$
Sign $0.894 - 0.447i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s i·3-s − 4-s + (−2 + i)5-s + 6-s i·8-s − 9-s + (−1 − 2i)10-s − 5·11-s + i·12-s i·13-s + (1 + 2i)15-s + 16-s − 2i·17-s i·18-s + 7·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.894 + 0.447i)5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + (−0.316 − 0.632i)10-s − 1.50·11-s + 0.288i·12-s − 0.277i·13-s + (0.258 + 0.516i)15-s + 0.250·16-s − 0.485i·17-s − 0.235i·18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.894 - 0.447i$
Motivic weight: \(1\)
Character: $\chi_{1470} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.088208770\)
\(L(\frac12)\) \(\approx\) \(1.088208770\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (2 - i)T \)
7 \( 1 \)
good11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 5iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 10iT - 43T^{2} \)
47 \( 1 + 13iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.506858341875896690407733367347, −8.324780208928235940542224213421, −7.79255165723859507121450273367, −7.35894510818848211701162857017, −6.47508992198743382441741548276, −5.44836009546409179146669427874, −4.78853758723080216506020188348, −3.43212948861950038759447224791, −2.65957645536533251781486224490, −0.71703134141693045798973665784, 0.75155539483712145679195271728, 2.48684151330262709961092652433, 3.38005002259004427887055595411, 4.32223120364492829701001689555, 5.01934241822842419254120534218, 5.83076933706513894768780898637, 7.36332304463235693496451944900, 7.966098742367669350050663928217, 8.753458223904739243849774299404, 9.523016099171869295072700801870

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