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Properties

Label 2-1470-105.104-c1-0-58
Degree $2$
Conductor $1470$
Sign $0.924 + 0.381i$
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 + (1.73 + 3.46i)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 + (0.448 + 0.893i)15-s + 0.250·16-s + 0.766i·17-s + (−0.471 − 0.527i)18-s + 0.328i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.924 + 0.381i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.924 + 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506843710\)
\(L(\frac12)\) \(\approx\) \(2.506843710\)
\(L(\frac{3}{2})\) 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

−9.523680395606912064660109802285, −8.722848823390086007579413327085, −8.072845364967814310477369188107, −6.57419845143869294830609390380, −5.74205107800904924058822871127, −5.52846468618743469210348480892, −4.33287211366824884852007995244, −3.72101810907084682765045100331, −2.53070511784759051123165775547, −0.889456567586859239066329556391, 1.46847700278065354273376826997, 2.41731649365692021992534308305, 3.30252083323582653005041356019, 4.86807202450722538489141807633, 5.32535991713365173393032853798, 6.50971342328689611885364553824, 6.93597899890110376021167987307, 7.45593520885631134650098463247, 8.696138738126471981763551176073, 9.702881455043905710998214375353

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