Properties

Label 2-1470-5.4-c1-0-34
Degree $2$
Conductor $1470$
Sign $-0.447 + 0.894i$
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 + (1 − 2i)5-s + 6-s + i·8-s − 9-s + (−2 − i)10-s i·12-s − 2i·13-s + (2 + i)15-s + 16-s + 2i·17-s + i·18-s + 2·19-s + (−1 + 2i)20-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.447 − 0.894i)5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + (−0.632 − 0.316i)10-s − 0.288i·12-s − 0.554i·13-s + (0.516 + 0.258i)15-s + 0.250·16-s + 0.485i·17-s + 0.235i·18-s + 0.458·19-s + (−0.223 + 0.447i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.447 + 0.894i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.444211781\)
\(L(\frac12)\) \(\approx\) \(1.444211781\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
7 \( 1 \)
good11 \( 1 + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 14iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−9.353434673950186741501825360433, −8.577952003610346816454468393426, −8.105252529058410679088420808538, −6.66147214651667964574699566764, −5.64468713834121442748398412585, −4.92719653795454403056502388287, −4.17220307708890667225788901902, −3.10495908572285584295515131038, −1.98376918962552250048975071727, −0.60111634040376251457823421806, 1.44465338010747811144219153142, 2.74580131967526383533619143421, 3.72258312326547493989756657378, 5.08203269684506091718479567691, 5.78841169345574841325169294100, 6.78407442677367540423210949802, 7.07290660704053498239680768188, 7.981977415397311781779893080535, 8.861454167459039808761036671625, 9.729200764796119309181695624812

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