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Properties

Label 2-1470-105.104-c1-0-21
Degree $2$
Conductor $1470$
Sign $0.928 + 0.372i$
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 + (−1.67 + 0.425i)3-s + 4-s + (−2.18 − 0.461i)5-s + (1.67 − 0.425i)6-s − 8-s + (2.63 − 1.42i)9-s + (2.18 + 0.461i)10-s + 4.08i·11-s + (−1.67 + 0.425i)12-s − 3.50·13-s + (3.86 − 0.155i)15-s + 16-s − 0.437i·17-s + (−2.63 + 1.42i)18-s − 7.69i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.969 + 0.245i)3-s + 0.5·4-s + (−0.978 − 0.206i)5-s + (0.685 − 0.173i)6-s − 0.353·8-s + (0.879 − 0.475i)9-s + (0.691 + 0.145i)10-s + 1.23i·11-s + (−0.484 + 0.122i)12-s − 0.970·13-s + (0.999 − 0.0401i)15-s + 0.250·16-s − 0.106i·17-s + (−0.621 + 0.336i)18-s − 1.76i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.928 + 0.372i$
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: \((2,\ 1470,\ (\ :1/2),\ 0.928 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4229961506\)
\(L(\frac12)\) \(\approx\) \(0.4229961506\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 0.425i)T \)
5 \( 1 + (2.18 + 0.461i)T \)
7 \( 1 \)
good11 \( 1 - 4.08iT - 11T^{2} \)
13 \( 1 + 3.50T + 13T^{2} \)
17 \( 1 + 0.437iT - 17T^{2} \)
19 \( 1 + 7.69iT - 19T^{2} \)
23 \( 1 + 7.39T + 23T^{2} \)
29 \( 1 - 4.95iT - 29T^{2} \)
31 \( 1 - 6.81iT - 31T^{2} \)
37 \( 1 - 4.51iT - 37T^{2} \)
41 \( 1 + 1.34T + 41T^{2} \)
43 \( 1 + 6.67iT - 43T^{2} \)
47 \( 1 + 2.41iT - 47T^{2} \)
53 \( 1 + 0.424T + 53T^{2} \)
59 \( 1 + 2.75T + 59T^{2} \)
61 \( 1 + 5.75iT - 61T^{2} \)
67 \( 1 + 11.3iT - 67T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 - 9.06T + 73T^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 - 17.1iT - 83T^{2} \)
89 \( 1 - 4.52T + 89T^{2} \)
97 \( 1 - 7.93T + 97T^{2} \)
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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

−9.542866436759522497183902712864, −8.776637032264833596807947282141, −7.67353715466285098935984155881, −7.10926640081467321495577561712, −6.50719193739663184342397482441, −4.98637655296152979681959084878, −4.73435310958877574251537431119, −3.46062660014753998142967883618, −1.98770282179482195506252307520, −0.43009971150199124708657657592, 0.64081856728867411922220667194, 2.14769961197423696241100294614, 3.58434580219535314184207078830, 4.44630832349643091387299423194, 5.85534112672663667033250016341, 6.14343185047166772235783063457, 7.43284224482476319980837741608, 7.80257556706932380911874914381, 8.527544038966599310286474797628, 9.859963969587236390543960383879

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