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Properties

Label 2-1470-35.4-c1-0-34
Degree $2$
Conductor $1470$
Sign $-0.523 + 0.851i$
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 + (1.40 − 1.74i)5-s − 0.999·6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.344 − 2.20i)10-s + (2.39 − 4.14i)11-s + (−0.866 + 0.499i)12-s − 3.17i·13-s + (−2.08 + 0.806i)15-s + (−0.5 − 0.866i)16-s + (4.52 + 2.61i)17-s + (0.866 + 0.499i)18-s + (1.58 + 2.74i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.627 − 0.778i)5-s − 0.408·6-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.108 − 0.698i)10-s + (0.721 − 1.24i)11-s + (−0.249 + 0.144i)12-s − 0.879i·13-s + (−0.538 + 0.208i)15-s + (−0.125 − 0.216i)16-s + (1.09 + 0.633i)17-s + (0.204 + 0.117i)18-s + (0.363 + 0.630i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.523 + 0.851i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.523 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.314227822\)
\(L(\frac12)\) \(\approx\) \(2.314227822\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 + 1.74i)T \)
7 \( 1 \)
good11 \( 1 + (-2.39 + 4.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.17iT - 13T^{2} \)
17 \( 1 + (-4.52 - 2.61i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.58 - 2.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.21 - 3.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.38T + 29T^{2} \)
31 \( 1 + (-2.08 + 3.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.21 + 3.58i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.05T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (9.97 - 5.75i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.02 + 4.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.36 + 9.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.55 - 6.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.28 + 4.78i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.08 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.39iT - 83T^{2} \)
89 \( 1 + (4.78 + 8.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.27iT - 97T^{2} \)
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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

−9.563724581291502396876448451119, −8.262236729262328031169153918524, −7.79374281990136053376576413050, −6.23329944797569383099453343892, −5.90286671219032975500041052964, −5.28995210751047783574218616805, −4.08960139873016470123274789969, −3.23163897162158366410207211042, −1.76790813997428965929235492897, −0.842135297937109811645707727774, 1.72006787850370349578719949155, 2.87024728096989388220074167501, 4.01746014720656015939342058397, 4.78592687835737357498911503732, 5.68769515230340526116496961072, 6.58975086843657690783108595297, 6.96432262342909582090639387842, 7.905598929055026945143983547851, 9.262407487378087822629686616130, 9.778244261829318373755723674330

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