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Properties

Label 2-1470-35.9-c1-0-29
Degree $2$
Conductor $1470$
Sign $0.330 + 0.943i$
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  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (2.23 + 0.133i)5-s − 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.86 − 1.23i)10-s + (0.866 + 0.499i)12-s − 2i·13-s + (1.99 − i)15-s + (−0.5 + 0.866i)16-s + (1.73 − i)17-s + (−0.866 + 0.499i)18-s + (1 − 1.73i)19-s + (0.999 + 1.99i)20-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (0.998 + 0.0599i)5-s − 0.408·6-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.590 − 0.389i)10-s + (0.249 + 0.144i)12-s − 0.554i·13-s + (0.516 − 0.258i)15-s + (−0.125 + 0.216i)16-s + (0.420 − 0.242i)17-s + (−0.204 + 0.117i)18-s + (0.229 − 0.397i)19-s + (0.223 + 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.330 + 0.943i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.330 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.789239948\)
\(L(\frac12)\) \(\approx\) \(1.789239948\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-2.23 - 0.133i)T \)
7 \( 1 \)
good11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.92 + 4i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.19 + 3i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.1 + 7i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.421428665126762593655057387327, −8.590876228387528245642487531064, −7.956918876610620780846254810477, −7.01726465407640459417086498725, −6.22700092173032153610153034138, −5.30729381166624692909130761414, −4.03128747738821136445894920105, −2.82519728467552799895120944827, −2.17096295884122059488048914414, −0.886516210025736502246084530831, 1.39957717309160398726180073344, 2.34149241273151373401002208591, 3.56594789470210554364709677015, 4.77539691794676714674716141375, 5.69811465275380996041629129026, 6.42361391667562408307510778561, 7.33994709373851324888819647748, 8.250543174455954149520500995999, 8.847818897895699102391366278643, 9.800127248936871008803469742596

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