Properties

Label 2-1620-45.29-c2-0-32
Degree $2$
Conductor $1620$
Sign $0.578 + 0.815i$
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  + (4.82 − 1.32i)5-s + (−5.42 − 3.13i)7-s + (0.895 + 0.516i)11-s + (2.59 − 1.5i)13-s − 15.7·17-s + 18.7·19-s + (17.3 + 30.0i)23-s + (21.5 − 12.7i)25-s + (2.68 + 1.55i)29-s + (8.89 + 15.4i)31-s + (−30.2 − 7.93i)35-s − 43.3i·37-s + (40.0 − 23.1i)41-s + (−17.9 − 10.3i)43-s + (39.4 − 68.3i)47-s + ⋯
L(s)  = 1  + (0.964 − 0.264i)5-s + (−0.774 − 0.447i)7-s + (0.0813 + 0.0469i)11-s + (0.199 − 0.115i)13-s − 0.928·17-s + 0.988·19-s + (0.753 + 1.30i)23-s + (0.860 − 0.509i)25-s + (0.0926 + 0.0534i)29-s + (0.286 + 0.496i)31-s + (−0.865 − 0.226i)35-s − 1.17i·37-s + (0.977 − 0.564i)41-s + (−0.417 − 0.241i)43-s + (0.839 − 1.45i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.578 + 0.815i$
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: \((2,\ 1620,\ (\ :1),\ 0.578 + 0.815i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.143674344\)
\(L(\frac12)\) \(\approx\) \(2.143674344\)
\(L(2)\) 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 \( 1 + (-4.82 + 1.32i)T \)
good7 \( 1 + (5.42 + 3.13i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.895 - 0.516i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.59 + 1.5i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 15.7T + 289T^{2} \)
19 \( 1 - 18.7T + 361T^{2} \)
23 \( 1 + (-17.3 - 30.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-2.68 - 1.55i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-8.89 - 15.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 43.3iT - 1.36e3T^{2} \)
41 \( 1 + (-40.0 + 23.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (17.9 + 10.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-39.4 + 68.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 44.2T + 2.80e3T^{2} \)
59 \( 1 + (-78.4 + 45.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (4.39 - 7.60i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-16.7 + 9.65i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 56.6iT - 5.04e3T^{2} \)
73 \( 1 - 109. iT - 5.32e3T^{2} \)
79 \( 1 + (19.5 - 33.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-37.8 + 65.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 90.5iT - 7.92e3T^{2} \)
97 \( 1 + (7.69 + 4.44i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.268212428455853909183475521401, −8.475218371405990786527252740629, −7.24998836200750324240268580385, −6.76188352975007508462842148125, −5.75469190610463016403590144538, −5.14312660260752853741989725520, −3.94935548499519385989512557157, −3.02221852784804493227849586568, −1.87010989916517708607386313876, −0.66389412791677015300529267444, 1.06505472719172310580654269798, 2.45931577642602450475206632029, 3.03783956838609571453874875315, 4.39247762786954040827573667983, 5.31645450569017722340509721421, 6.35010117236854479836088709685, 6.54458535080308877182907496290, 7.71060330076093661146809734728, 8.802133916193991392585505989417, 9.326247722198558139757959728394

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