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Properties

Label 2-1470-35.9-c1-0-34
Degree $2$
Conductor $1470$
Sign $0.220 + 0.975i$
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 + (−2.13 + 0.650i)5-s + 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.17 − 0.506i)10-s + (−2.23 − 3.87i)11-s + (0.866 + 0.499i)12-s − 5.88i·13-s + (−1.52 + 1.63i)15-s + (−0.5 + 0.866i)16-s + (−6.69 + 3.86i)17-s + (0.866 − 0.499i)18-s + (3.30 − 5.73i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.956 + 0.290i)5-s + 0.408·6-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.688 − 0.160i)10-s + (−0.673 − 1.16i)11-s + (0.249 + 0.144i)12-s − 1.63i·13-s + (−0.394 + 0.421i)15-s + (−0.125 + 0.216i)16-s + (−1.62 + 0.938i)17-s + (0.204 − 0.117i)18-s + (0.759 − 1.31i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.220 + 0.975i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.699553450\)
\(L(\frac12)\) \(\approx\) \(1.699553450\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 - 0.650i)T \)
7 \( 1 \)
good11 \( 1 + (2.23 + 3.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.88iT - 13T^{2} \)
17 \( 1 + (6.69 - 3.86i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.30 + 5.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.26 - 1.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.17T + 29T^{2} \)
31 \( 1 + (4.23 + 7.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.76 + 1.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.56T + 41T^{2} \)
43 \( 1 + 1.43iT - 43T^{2} \)
47 \( 1 + (5.88 + 3.39i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.19 + 0.690i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.33 - 4.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.89 - 8.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.60 + 0.925i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.02T + 71T^{2} \)
73 \( 1 + (3.47 - 2.00i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.49 - 6.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.35iT - 83T^{2} \)
89 \( 1 + (-7.19 + 12.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.71iT - 97T^{2} \)
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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

−8.900037054696774421104858382287, −8.384053231209082472874644529855, −7.70575628666276615209094462288, −6.99813566083998640468614209758, −6.08144132983209814694723263694, −5.16216347976982468771876570745, −4.15184499116948688612281910591, −3.17230344404762760706101594482, −2.64005154940745889155754090105, −0.50312789182697152461837699072, 1.66362377236020014000786653145, 2.73651002541476139972725977332, 3.78975406867242080824409443007, 4.68869259989233303562592240712, 4.92129724808190579148204638045, 6.64012562988585466261412921655, 7.14125103578300582133560142613, 8.094139634003302539474680769485, 9.009344306207209739913638981917, 9.602507890144611758990989746501

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