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Properties

Label 2-1470-105.104-c1-0-14
Degree $2$
Conductor $1470$
Sign $-0.558 - 0.829i$
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 + (0.707 + 1.58i)3-s + 4-s + (1.41 − 1.73i)5-s + (−0.707 − 1.58i)6-s − 8-s + (−2.00 + 2.23i)9-s + (−1.41 + 1.73i)10-s + 4.68i·11-s + (0.707 + 1.58i)12-s − 1.04·13-s + (3.73 + 1.01i)15-s + 16-s + 3.16i·17-s + (2.00 − 2.23i)18-s − 1.43i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 + 0.912i)3-s + 0.5·4-s + (0.632 − 0.774i)5-s + (−0.288 − 0.645i)6-s − 0.353·8-s + (−0.666 + 0.745i)9-s + (−0.447 + 0.547i)10-s + 1.41i·11-s + (0.204 + 0.456i)12-s − 0.289·13-s + (0.965 + 0.261i)15-s + 0.250·16-s + 0.766i·17-s + (0.471 − 0.527i)18-s − 0.328i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.558 - 0.829i$
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: \((2,\ 1470,\ (\ :1/2),\ -0.558 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.144662510\)
\(L(\frac12)\) \(\approx\) \(1.144662510\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 1.58i)T \)
5 \( 1 + (-1.41 + 1.73i)T \)
7 \( 1 \)
good11 \( 1 - 4.68iT - 11T^{2} \)
13 \( 1 + 1.04T + 13T^{2} \)
17 \( 1 - 3.16iT - 17T^{2} \)
19 \( 1 + 1.43iT - 19T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 6.62iT - 31T^{2} \)
37 \( 1 + 2.66iT - 37T^{2} \)
41 \( 1 + 1.04T + 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 - 11.2iT - 47T^{2} \)
53 \( 1 + 5T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 14.2iT - 67T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 - 3.50T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 4.06iT - 83T^{2} \)
89 \( 1 - 4.91T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−9.720551024822547576968172256307, −9.076383260695924543053921943661, −8.464411987237581576271583477625, −7.61134636085366999699225158684, −6.62934791858120858346199269569, −5.49751024179264339124322320275, −4.79747598336732659991890513091, −3.87330205438505901124272572285, −2.50242728913624812642994384123, −1.61532086531485661076429697799, 0.51846805883156375540553485388, 1.96314945289196865808231244369, 2.75070532990502610782105288628, 3.66029344571901806020527654636, 5.52748517766612541413645857937, 6.22577529970691988018086343566, 6.79725961412839819740127111819, 7.86224047199784813754740335570, 8.185151302310617427450502364460, 9.317509417598013488643556306450

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