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Properties

Label 2-1470-5.4-c1-0-18
Degree $2$
Conductor $1470$
Sign $0.894 + 0.447i$
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  − i·2-s − i·3-s − 4-s + (2 + i)5-s − 6-s + i·8-s − 9-s + (1 − 2i)10-s + 2·11-s + i·12-s + 6i·13-s + (1 − 2i)15-s + 16-s − 2i·17-s + i·18-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 + 0.603·11-s + 0.288i·12-s + 1.66i·13-s + (0.258 − 0.516i)15-s + 0.250·16-s − 0.485i·17-s + 0.235i·18-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: $0.894 + 0.447i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
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

\(L(1)\) \(\approx\) \(1.891417695\)
\(L(\frac12)\) \(\approx\) \(1.891417695\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
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

−9.477712427087874597735074880976, −8.955552114257713804099780591386, −7.88094156907897375622247621512, −6.77572921875698048847123232792, −6.39644751423259458431223566533, −5.26316828814816728971688627562, −4.26985893026235069627115502598, −3.11398472898885359095516035133, −2.12487353228441756462978713389, −1.30261018380362140928162292080, 0.865571406357834911579609005348, 2.53322819024110257694410010583, 3.73917199616717111911288681286, 4.74496370413980754209600536122, 5.51024514190473386978132584240, 6.12271611463750799361807268494, 7.01389206298808619852366448788, 8.313272395432225578297641864524, 8.534156466492753092464422334564, 9.593148554494952185696762801018

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