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Properties

Label 2-1470-35.4-c1-0-1
Degree $2$
Conductor $1470$
Sign $-0.814 + 0.580i$
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 + (−0.506 + 2.17i)5-s − 0.999·6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.650 − 2.13i)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 + (−0.226 + 0.973i)5-s − 0.408·6-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.205 − 0.676i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.814 + 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4948654749\)
\(L(\frac12)\) \(\approx\) \(0.4948654749\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.506 - 2.17i)T \)
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

−9.813479250435535741456032768919, −9.231093352258552502973690357142, −8.451276887515895556407381335251, −7.47676563559933234896084140642, −6.86619361217379769991065378060, −6.37707846070265472192193019371, −4.67827133099322886008805686141, −4.32528943310738346956276193924, −2.59548635627968857866093227758, −2.19493338618892084401510407904, 0.21670373252245223976623442108, 1.46481558855433681448129164952, 2.71142976092104606869461216577, 3.63126021369440837315991505654, 4.70058178172154784505301199355, 5.83031032277726262290395485306, 6.62532553438582119350460333350, 8.068427970716985454205587718024, 8.346578280289303257158104671289, 8.556625602109833083038653002612

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