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Properties

Label 2-1620-45.14-c2-0-8
Degree $2$
Conductor $1620$
Sign $-0.481 - 0.876i$
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  + (4.98 − 0.332i)5-s + (−0.607 + 0.350i)7-s + (−0.400 + 0.231i)11-s + (−12.2 − 7.05i)13-s − 14.3·17-s + 5.37·19-s + (−20.8 + 36.0i)23-s + (24.7 − 3.31i)25-s + (−8.09 + 4.67i)29-s + (−10.9 + 18.9i)31-s + (−2.91 + 1.95i)35-s + 34.0i·37-s + (19.3 + 11.1i)41-s + (−48.4 + 27.9i)43-s + (−3.82 − 6.62i)47-s + ⋯
L(s)  = 1  + (0.997 − 0.0664i)5-s + (−0.0868 + 0.0501i)7-s + (−0.0364 + 0.0210i)11-s + (−0.939 − 0.542i)13-s − 0.846·17-s + 0.283·19-s + (−0.905 + 1.56i)23-s + (0.991 − 0.132i)25-s + (−0.279 + 0.161i)29-s + (−0.352 + 0.611i)31-s + (−0.0833 + 0.0558i)35-s + 0.921i·37-s + (0.472 + 0.272i)41-s + (−1.12 + 0.650i)43-s + (−0.0813 − 0.140i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.481 - 0.876i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.124474130\)
\(L(\frac12)\) \(\approx\) \(1.124474130\)
\(L(2)\) 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 \( 1 + (-4.98 + 0.332i)T \)
good7 \( 1 + (0.607 - 0.350i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (0.400 - 0.231i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (12.2 + 7.05i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 14.3T + 289T^{2} \)
19 \( 1 - 5.37T + 361T^{2} \)
23 \( 1 + (20.8 - 36.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (8.09 - 4.67i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (10.9 - 18.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 34.0iT - 1.36e3T^{2} \)
41 \( 1 + (-19.3 - 11.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (48.4 - 27.9i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (3.82 + 6.62i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 36.2T + 2.80e3T^{2} \)
59 \( 1 + (-30.3 - 17.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-29.4 - 50.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (53.6 + 30.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 43.6iT - 5.04e3T^{2} \)
73 \( 1 + 50.7iT - 5.32e3T^{2} \)
79 \( 1 + (-48.0 - 83.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-13.0 - 22.5i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 125. iT - 7.92e3T^{2} \)
97 \( 1 + (102. - 58.8i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.614385301123503420512246987672, −8.804693286472158213351778790747, −7.85049301746265730620056194642, −7.04800221915221667890354745688, −6.16672935860369116910261515761, −5.39805262178969367887544603302, −4.67982745468259187404173476850, −3.37446525792773504078232548485, −2.40023560444466509510309786589, −1.39925251791206547652349950361, 0.27755741325112250590378261486, 1.95472286858664224619487097221, 2.52476337833736958755536852884, 3.93994000028513211527637695607, 4.84798209643901544947920349427, 5.70211596578278334776048244784, 6.57221641346481359587596233911, 7.15050119258427380423710910698, 8.274713295037244293991697882664, 9.047941017999821450772650015152

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