Properties

Label 2-1470-5.4-c1-0-17
Degree $2$
Conductor $1470$
Sign $0.975 - 0.218i$
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 + (2.18 − 0.489i)5-s + 6-s + i·8-s − 9-s + (−0.489 − 2.18i)10-s − 2.39·11-s i·12-s + 3.80i·13-s + (0.489 + 2.18i)15-s + 16-s − 1.97i·17-s + i·18-s + 4.17·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.975 − 0.218i)5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + (−0.154 − 0.689i)10-s − 0.721·11-s − 0.288i·12-s + 1.05i·13-s + (0.126 + 0.563i)15-s + 0.250·16-s − 0.477i·17-s + 0.235i·18-s + 0.956·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.975 - 0.218i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.798724472\)
\(L(\frac12)\) \(\approx\) \(1.798724472\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 + 0.489i)T \)
7 \( 1 \)
good11 \( 1 + 2.39T + 11T^{2} \)
13 \( 1 - 3.80iT - 13T^{2} \)
17 \( 1 + 1.97iT - 17T^{2} \)
19 \( 1 - 4.17T + 19T^{2} \)
23 \( 1 - 8.17iT - 23T^{2} \)
29 \( 1 - 9.16T + 29T^{2} \)
31 \( 1 - 1.60T + 31T^{2} \)
37 \( 1 + 1.26iT - 37T^{2} \)
41 \( 1 + 3.19T + 41T^{2} \)
43 \( 1 - 4.90iT - 43T^{2} \)
47 \( 1 + 3.00iT - 47T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 - 5.77iT - 67T^{2} \)
71 \( 1 - 6.94T + 71T^{2} \)
73 \( 1 + 0.506iT - 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + 1.89iT - 83T^{2} \)
89 \( 1 + 1.97T + 89T^{2} \)
97 \( 1 + 12.0iT - 97T^{2} \)
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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

−9.617360607733013218036462452564, −9.119129794542891095215280994232, −8.188261562468839413553655041426, −7.11700294719672871858787230921, −6.00817997828253780348004983759, −5.17685270482764594268797402663, −4.56259205733786911895368571223, −3.32537994755828334772995216910, −2.45248716681999555701346070026, −1.25980443879619904709610678984, 0.817165491894354347360609516565, 2.34196292055494742292842965491, 3.23791243839408735456391988555, 4.84864937670624148621634675298, 5.44826871742068999035526542829, 6.36104640112555564255154908233, 6.84282922869384176055903669314, 8.000121770423606247170117463145, 8.349030041090431254435578112473, 9.408583362345584851969516023561

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