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Properties

Label 2-1470-35.9-c1-0-2
Degree $2$
Conductor $1470$
Sign $-0.398 - 0.917i$
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  + (−0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.98 + 1.03i)5-s − 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (2.23 + 0.0918i)10-s + (−0.489 − 0.847i)11-s + (0.866 + 0.499i)12-s + 0.435i·13-s + (−1.19 + 1.88i)15-s + (−0.5 + 0.866i)16-s + (2.42 − 1.39i)17-s + (−0.866 + 0.499i)18-s + (−3.67 + 6.35i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.885 + 0.464i)5-s − 0.408·6-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.706 + 0.0290i)10-s + (−0.147 − 0.255i)11-s + (0.249 + 0.144i)12-s + 0.120i·13-s + (−0.308 + 0.487i)15-s + (−0.125 + 0.216i)16-s + (0.587 − 0.339i)17-s + (−0.204 + 0.117i)18-s + (−0.842 + 1.45i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.398 - 0.917i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.398 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4471636180\)
\(L(\frac12)\) \(\approx\) \(0.4471636180\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (1.98 - 1.03i)T \)
7 \( 1 \)
good11 \( 1 + (0.489 + 0.847i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.435iT - 13T^{2} \)
17 \( 1 + (-2.42 + 1.39i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.67 - 6.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.89 + 1.67i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.74T + 29T^{2} \)
31 \( 1 + (-2.48 - 4.31i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.01 + 2.31i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.94T + 41T^{2} \)
43 \( 1 - 9.97iT - 43T^{2} \)
47 \( 1 + (-3.81 - 2.20i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.569 - 0.328i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.13 + 7.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.66 - 9.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.04 + 1.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + (13.3 - 7.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.44 - 2.50i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.3iT - 83T^{2} \)
89 \( 1 + (5.06 - 8.76i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−9.835421452176050492388583938650, −8.772771799963062069102422876495, −8.169529976621003907214622086034, −7.60817766021217318348437034491, −6.78896368201644098718108159132, −5.86589111415805673460477162657, −4.37108942105874482690963884041, −3.54388906770338420555397937724, −2.72108292546497674087842427235, −1.46932070033836511264978468800, 0.20488375368545718493750525164, 1.83499227751362700669276366318, 3.14692824897315495540153983692, 4.19701986198915072438413167995, 4.98885660091489626118871824317, 6.04544022067812615338168769481, 7.23142214447834133135617977001, 7.64594067954220864312137357574, 8.662308273266974712720813147646, 8.900935522527083566974184950091

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