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Properties

Label 2-1470-105.104-c1-0-75
Degree $2$
Conductor $1470$
Sign $-0.983 + 0.182i$
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  − 2-s + (1.17 − 1.27i)3-s + 4-s + (−1.07 − 1.95i)5-s + (−1.17 + 1.27i)6-s − 8-s + (−0.259 − 2.98i)9-s + (1.07 + 1.95i)10-s − 6.01i·11-s + (1.17 − 1.27i)12-s + 0.864·13-s + (−3.76 − 0.914i)15-s + 16-s + 2.84i·17-s + (0.259 + 2.98i)18-s − 7.12i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.675 − 0.737i)3-s + 0.5·4-s + (−0.482 − 0.875i)5-s + (−0.477 + 0.521i)6-s − 0.353·8-s + (−0.0865 − 0.996i)9-s + (0.341 + 0.619i)10-s − 1.81i·11-s + (0.337 − 0.368i)12-s + 0.239·13-s + (−0.971 − 0.236i)15-s + 0.250·16-s + 0.690i·17-s + (0.0612 + 0.704i)18-s − 1.63i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.983 + 0.182i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.100978771\)
\(L(\frac12)\) \(\approx\) \(1.100978771\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.17 + 1.27i)T \)
5 \( 1 + (1.07 + 1.95i)T \)
7 \( 1 \)
good11 \( 1 + 6.01iT - 11T^{2} \)
13 \( 1 - 0.864T + 13T^{2} \)
17 \( 1 - 2.84iT - 17T^{2} \)
19 \( 1 + 7.12iT - 19T^{2} \)
23 \( 1 - 2.10T + 23T^{2} \)
29 \( 1 - 9.40iT - 29T^{2} \)
31 \( 1 + 2.68iT - 31T^{2} \)
37 \( 1 - 6.81iT - 37T^{2} \)
41 \( 1 - 5.32T + 41T^{2} \)
43 \( 1 + 2.68iT - 43T^{2} \)
47 \( 1 + 3.12iT - 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + 7.09T + 59T^{2} \)
61 \( 1 + 9.33iT - 61T^{2} \)
67 \( 1 - 6.56iT - 67T^{2} \)
71 \( 1 + 4.46iT - 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 3.35T + 79T^{2} \)
83 \( 1 + 1.28iT - 83T^{2} \)
89 \( 1 + 3.08T + 89T^{2} \)
97 \( 1 - 1.90T + 97T^{2} \)
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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

−8.892377935847161539903152779450, −8.436110842658307759760550859079, −7.82740963619674363547238756297, −6.85626671298525285509191828385, −6.10410180199232433082290779110, −5.01760552719573656393243975124, −3.62843283845936447615745957360, −2.90464674268220467217586232528, −1.43403832149458729566738971802, −0.52162290362040432893778074515, 1.91006867373459928363812471432, 2.78338256826195865094911668666, 3.85377694325974888749849235711, 4.60387483175348795145379882591, 5.90937931612070332833771657366, 6.98331495625899944490843927689, 7.71404031462592127728127253453, 8.101295982830723147110304807043, 9.448995564451372328799026272506, 9.658612887997822969232427819172

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