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Properties

Label 2-1470-35.9-c1-0-28
Degree $2$
Conductor $1470$
Sign $0.981 - 0.192i$
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 + (−1 − 1.73i)11-s + (−0.866 − 0.499i)12-s − 6i·13-s + (−1.99 + i)15-s + (−0.5 + 0.866i)16-s + (3.46 − 2i)17-s + (0.866 − 0.499i)18-s + (3 − 5.19i)19-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 + (−0.301 − 0.522i)11-s + (−0.249 − 0.144i)12-s − 1.66i·13-s + (−0.516 + 0.258i)15-s + (−0.125 + 0.216i)16-s + (0.840 − 0.485i)17-s + (0.204 − 0.117i)18-s + (0.688 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.981 - 0.192i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.981 - 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.511332560\)
\(L(\frac12)\) \(\approx\) \(2.511332560\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.92 - 4i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (6.92 + 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-12.1 + 7i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8iT - 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−9.610773095498439021277484145794, −8.804224451387624610966165774247, −7.66796995502018494963124604814, −7.02888426154875986987259831919, −5.90483061202324443734226714843, −5.41704022637480138333897441283, −4.92771131404841427189294386469, −3.35546523042831410493943271320, −2.78330481029868093181720790861, −0.984692591527095093755742029180, 1.39794352721693562270798776356, 2.11489991981014125195562391935, 3.42953368669695346740675169272, 4.59554859236446153826338688743, 5.31383171730744051321805410474, 6.10811633797314390263028127198, 6.80300890410985513966541752420, 7.63598803461106712792353480062, 8.937233406824895340214914029973, 9.638067468697833204099021291143

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