Properties

Label 2-1470-5.4-c1-0-23
Degree $2$
Conductor $1470$
Sign $0.627 - 0.778i$
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 + (1.40 − 1.74i)5-s − 6-s i·8-s − 9-s + (1.74 + 1.40i)10-s + 1.67·11-s i·12-s + 4.48i·13-s + (1.74 + 1.40i)15-s + 16-s − 7.61i·17-s i·18-s + 4.48·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.627 − 0.778i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.550 + 0.443i)10-s + 0.505·11-s − 0.288i·12-s + 1.24i·13-s + (0.449 + 0.362i)15-s + 0.250·16-s − 1.84i·17-s − 0.235i·18-s + 1.02·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.627 - 0.778i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.627 - 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.863904000\)
\(L(\frac12)\) \(\approx\) \(1.863904000\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 1.74i)T \)
7 \( 1 \)
good11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 - 4.48iT - 13T^{2} \)
17 \( 1 + 7.61iT - 17T^{2} \)
19 \( 1 - 4.48T + 19T^{2} \)
23 \( 1 - 0.482iT - 23T^{2} \)
29 \( 1 + 1.19T + 29T^{2} \)
31 \( 1 - 3.48T + 31T^{2} \)
37 \( 1 + 0.482iT - 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 9.57iT - 53T^{2} \)
59 \( 1 - 5.77T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + 3.35iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 4.51T + 79T^{2} \)
83 \( 1 - 1.87iT - 83T^{2} \)
89 \( 1 + 3.35T + 89T^{2} \)
97 \( 1 - 17.7iT - 97T^{2} \)
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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

−9.369031856481643888005508266903, −9.134814103096724052938441472349, −8.060927085670340726748104766062, −7.12674402835873759033735402680, −6.32045250165647799624219134508, −5.37590887083313327672320317452, −4.76584072817582689844934517498, −3.95994226688819393580873393130, −2.56240057973805297866905676386, −0.990336323856238504529014145970, 1.07991343466882245388602156908, 2.18176813408522913471207217272, 3.12125916206980249582683792413, 3.97883123421067713442361256920, 5.48283828967980429456917003308, 5.98141244023662227906757248540, 6.96418055541242475640467519513, 7.85636271804100105540403561558, 8.607499852350410162753531810577, 9.591862912525457049249391236699

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