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Properties

Label 2-1470-7.6-c2-0-19
Degree $2$
Conductor $1470$
Sign $0.755 - 0.654i$
Analytic cond. $40.0545$
Root an. cond. $6.32887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s − 1.73i·3-s + 2.00·4-s + 2.23i·5-s − 2.44i·6-s + 2.82·8-s − 2.99·9-s + 3.16i·10-s − 17.3·11-s − 3.46i·12-s + 7.22i·13-s + 3.87·15-s + 4.00·16-s − 2.65i·17-s − 4.24·18-s − 2.54i·19-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.447i·5-s − 0.408i·6-s + 0.353·8-s − 0.333·9-s + 0.316i·10-s − 1.58·11-s − 0.288i·12-s + 0.555i·13-s + 0.258·15-s + 0.250·16-s − 0.155i·17-s − 0.235·18-s − 0.133i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(40.0545\)
Root analytic conductor: \(6.32887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.631904654\)
\(L(\frac12)\) \(\approx\) \(2.631904654\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + 1.73iT \)
5 \( 1 - 2.23iT \)
7 \( 1 \)
good11 \( 1 + 17.3T + 121T^{2} \)
13 \( 1 - 7.22iT - 169T^{2} \)
17 \( 1 + 2.65iT - 289T^{2} \)
19 \( 1 + 2.54iT - 361T^{2} \)
23 \( 1 - 40.1T + 529T^{2} \)
29 \( 1 - 47.0T + 841T^{2} \)
31 \( 1 - 40.3iT - 961T^{2} \)
37 \( 1 - 32.5T + 1.36e3T^{2} \)
41 \( 1 - 70.6iT - 1.68e3T^{2} \)
43 \( 1 - 37.3T + 1.84e3T^{2} \)
47 \( 1 - 33.4iT - 2.20e3T^{2} \)
53 \( 1 + 70.8T + 2.80e3T^{2} \)
59 \( 1 - 100. iT - 3.48e3T^{2} \)
61 \( 1 - 12.8iT - 3.72e3T^{2} \)
67 \( 1 - 94.0T + 4.48e3T^{2} \)
71 \( 1 + 11.5T + 5.04e3T^{2} \)
73 \( 1 + 22.6iT - 5.32e3T^{2} \)
79 \( 1 + 24.1T + 6.24e3T^{2} \)
83 \( 1 + 111. iT - 6.88e3T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 - 7.48iT - 9.40e3T^{2} \)
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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

−9.431171776132721283268304207663, −8.380188584505418734445197963151, −7.62821467557639609545473397929, −6.89220229518465934094324821545, −6.21584608808691043991623282697, −5.17271678206775296968162798070, −4.54643596733327183638110545945, −2.97570084507025159198502014253, −2.67616816603943221207130833791, −1.16950023210478508936444514454, 0.61541795686309939793120018802, 2.37816163700860833644525908542, 3.15648897810064336596223722961, 4.26978163121635107422592864265, 5.11571058263909351255438381844, 5.54144356897033286769002377419, 6.64161205472088123701933877868, 7.73107453001411124576451211850, 8.292310102539877390031836066006, 9.308419368661283005706000316795

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