Properties

Label 2-1620-45.29-c2-0-2
Degree $2$
Conductor $1620$
Sign $-0.421 - 0.906i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + (2.11 − 4.52i)5-s + (−6.19 − 3.57i)7-s + (−4.39 − 2.53i)11-s + (2.70 − 1.56i)13-s − 8.72·17-s − 20.1·19-s + (−7.36 − 12.7i)23-s + (−16.0 − 19.1i)25-s + (34.4 + 19.8i)29-s + (19.6 + 34.0i)31-s + (−29.3 + 20.4i)35-s + 34.8i·37-s + (−11.4 + 6.63i)41-s + (−57.6 − 33.3i)43-s + (−8.45 + 14.6i)47-s + ⋯
L(s)  = 1  + (0.423 − 0.905i)5-s + (−0.885 − 0.511i)7-s + (−0.399 − 0.230i)11-s + (0.208 − 0.120i)13-s − 0.513·17-s − 1.06·19-s + (−0.320 − 0.554i)23-s + (−0.641 − 0.767i)25-s + (1.18 + 0.686i)29-s + (0.635 + 1.09i)31-s + (−0.837 + 0.585i)35-s + 0.942i·37-s + (−0.280 + 0.161i)41-s + (−1.34 − 0.774i)43-s + (−0.179 + 0.311i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.421 - 0.906i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.421 - 0.906i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1989954710\)
\(L(\frac12)\) \(\approx\) \(0.1989954710\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.11 + 4.52i)T \)
good7 \( 1 + (6.19 + 3.57i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.39 + 2.53i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.70 + 1.56i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 8.72T + 289T^{2} \)
19 \( 1 + 20.1T + 361T^{2} \)
23 \( 1 + (7.36 + 12.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-34.4 - 19.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-19.6 - 34.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 34.8iT - 1.36e3T^{2} \)
41 \( 1 + (11.4 - 6.63i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (57.6 + 33.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (8.45 - 14.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 4.62T + 2.80e3T^{2} \)
59 \( 1 + (22.3 - 12.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-6.09 + 10.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-92.1 + 53.2i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 101. iT - 5.04e3T^{2} \)
73 \( 1 - 23.2iT - 5.32e3T^{2} \)
79 \( 1 + (-33.2 + 57.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (72.1 - 124. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 154. iT - 7.92e3T^{2} \)
97 \( 1 + (-152. - 87.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.490460007441595246114808148805, −8.487404177250751141643689962114, −8.259629249769735303721016719700, −6.70217868861002018147771705607, −6.48115770297260816055374586243, −5.26501652651726388340955596389, −4.56034701244226190123994433145, −3.53862448186741028213085489403, −2.43764850633398577759564245037, −1.12589135929034622325486357764, 0.05509878630055200340151081481, 2.01265397277847014512684585265, 2.74219356938911570417820462814, 3.71427235244043478147694328730, 4.80617860161510754311328558428, 6.10915249806110121886196817213, 6.27498692662324328235836389496, 7.24426502583799186460228551103, 8.176878065908309050337668795257, 9.049019265350141335572749705306

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