Properties

Label 2-1470-35.13-c1-0-15
Degree $2$
Conductor $1470$
Sign $0.919 - 0.392i$
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  + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (2.17 − 0.537i)5-s − 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−1.15 + 1.91i)10-s − 1.16·11-s + (0.707 + 0.707i)12-s + (1.92 − 1.92i)13-s + (−1.15 + 1.91i)15-s − 1.00·16-s + (−0.0153 − 0.0153i)17-s + (0.707 + 0.707i)18-s − 1.97·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.970 − 0.240i)5-s − 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (−0.365 + 0.605i)10-s − 0.351·11-s + (0.204 + 0.204i)12-s + (0.533 − 0.533i)13-s + (−0.298 + 0.494i)15-s − 0.250·16-s + (−0.00371 − 0.00371i)17-s + (0.166 + 0.166i)18-s − 0.453·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.919 - 0.392i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.919 - 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.343269904\)
\(L(\frac12)\) \(\approx\) \(1.343269904\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-2.17 + 0.537i)T \)
7 \( 1 \)
good11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 + (-1.92 + 1.92i)T - 13iT^{2} \)
17 \( 1 + (0.0153 + 0.0153i)T + 17iT^{2} \)
19 \( 1 + 1.97T + 19T^{2} \)
23 \( 1 + (-5.07 - 5.07i)T + 23iT^{2} \)
29 \( 1 + 5.60iT - 29T^{2} \)
31 \( 1 - 7.93iT - 31T^{2} \)
37 \( 1 + (-7.51 + 7.51i)T - 37iT^{2} \)
41 \( 1 + 2.48iT - 41T^{2} \)
43 \( 1 + (7.87 + 7.87i)T + 43iT^{2} \)
47 \( 1 + (-2.88 - 2.88i)T + 47iT^{2} \)
53 \( 1 + (2.06 + 2.06i)T + 53iT^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 - 3.64iT - 61T^{2} \)
67 \( 1 + (-10.2 + 10.2i)T - 67iT^{2} \)
71 \( 1 - 7.51T + 71T^{2} \)
73 \( 1 + (2.64 - 2.64i)T - 73iT^{2} \)
79 \( 1 + 1.61iT - 79T^{2} \)
83 \( 1 + (-9.74 + 9.74i)T - 83iT^{2} \)
89 \( 1 - 3.60T + 89T^{2} \)
97 \( 1 + (-0.265 - 0.265i)T + 97iT^{2} \)
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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

−9.469152578175939393266807417293, −8.901649311010949202101045446821, −8.058683634773838366034941790218, −7.04184626727175244564503203866, −6.22214905939142017958484153420, −5.48693831002134688735948186209, −4.91630264850068465453710390099, −3.57968634243810928374018051657, −2.21217018657132541707858631242, −0.863087672066413038240605541738, 1.03168803028365219899622631855, 2.13348085638893930072250483428, 3.00643109150863393282802391353, 4.41106171527099424025334219720, 5.37011052524843439989544379343, 6.43615384607587407540910195183, 6.82631374465349865235177053148, 8.018337376263526691645410878045, 8.715814665049442371173378322384, 9.600195764717380813505279241067

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