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Properties

Label 2-1470-7.6-c2-0-15
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 + 19.9·11-s − 3.46i·12-s − 3.49i·13-s − 3.87·15-s + 4.00·16-s + 18.2i·17-s + 4.24·18-s + 24.6i·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 + 1.81·11-s − 0.288i·12-s − 0.269i·13-s − 0.258·15-s + 0.250·16-s + 1.07i·17-s + 0.235·18-s + 1.29i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.237225975\)
\(L(\frac12)\) \(\approx\) \(1.237225975\)
\(L(2)\) 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 \( 1 - 19.9T + 121T^{2} \)
13 \( 1 + 3.49iT - 169T^{2} \)
17 \( 1 - 18.2iT - 289T^{2} \)
19 \( 1 - 24.6iT - 361T^{2} \)
23 \( 1 - 25.1T + 529T^{2} \)
29 \( 1 + 53.1T + 841T^{2} \)
31 \( 1 - 30.1iT - 961T^{2} \)
37 \( 1 + 46.7T + 1.36e3T^{2} \)
41 \( 1 - 31.5iT - 1.68e3T^{2} \)
43 \( 1 - 64.4T + 1.84e3T^{2} \)
47 \( 1 - 28.0iT - 2.20e3T^{2} \)
53 \( 1 + 64.8T + 2.80e3T^{2} \)
59 \( 1 - 100. iT - 3.48e3T^{2} \)
61 \( 1 + 8.02iT - 3.72e3T^{2} \)
67 \( 1 - 16.2T + 4.48e3T^{2} \)
71 \( 1 + 107.T + 5.04e3T^{2} \)
73 \( 1 - 51.6iT - 5.32e3T^{2} \)
79 \( 1 + 21.9T + 6.24e3T^{2} \)
83 \( 1 + 0.417iT - 6.88e3T^{2} \)
89 \( 1 - 111. iT - 7.92e3T^{2} \)
97 \( 1 + 74.2iT - 9.40e3T^{2} \)
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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

−9.109073918689748720545857876014, −8.790655948674784977362871176836, −7.84291221944740078251568206819, −7.10581385713056168176776635734, −6.25215054531883601524284063202, −5.60153242944552983711078550300, −4.16931619814432965530393682696, −3.29808031587811914354096175036, −1.69619140495248695452146862712, −1.20675672109701065865316887219, 0.47656891515325075779950293950, 1.90570344498819671359158200341, 3.10958427768927564160478025398, 3.98610985788300107273059220526, 5.04144790716529527679693480427, 6.13900949777448545448737405317, 6.97664927851317893697834917026, 7.45935574571234350216420330734, 8.951790188548386805039479726239, 9.109119582956482776021516819426

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