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Properties

Label 2-1620-9.5-c2-0-3
Degree $2$
Conductor $1620$
Sign $-0.996 - 0.0871i$
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  + (−1.93 − 1.11i)5-s + (3.57 + 6.18i)7-s + (2.20 − 1.27i)11-s + (−5.57 + 9.66i)13-s − 2.05i·17-s − 19.3·19-s + (10.4 + 6.01i)23-s + (2.5 + 4.33i)25-s + (19.3 − 11.1i)29-s + (−7.08 + 12.2i)31-s − 15.9i·35-s − 46.3·37-s + (−15.0 − 8.70i)41-s + (−13.5 − 23.3i)43-s + (−42.2 + 24.3i)47-s + ⋯
L(s)  = 1  + (−0.387 − 0.223i)5-s + (0.510 + 0.884i)7-s + (0.200 − 0.115i)11-s + (−0.429 + 0.743i)13-s − 0.121i·17-s − 1.01·19-s + (0.453 + 0.261i)23-s + (0.100 + 0.173i)25-s + (0.665 − 0.384i)29-s + (−0.228 + 0.395i)31-s − 0.456i·35-s − 1.25·37-s + (−0.367 − 0.212i)41-s + (−0.314 − 0.544i)43-s + (−0.898 + 0.519i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.996 - 0.0871i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.996 - 0.0871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3839142994\)
\(L(\frac12)\) \(\approx\) \(0.3839142994\)
\(L(2)\) 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 \( 1 + (1.93 + 1.11i)T \)
good7 \( 1 + (-3.57 - 6.18i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-2.20 + 1.27i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5.57 - 9.66i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 2.05iT - 289T^{2} \)
19 \( 1 + 19.3T + 361T^{2} \)
23 \( 1 + (-10.4 - 6.01i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-19.3 + 11.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (7.08 - 12.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 46.3T + 1.36e3T^{2} \)
41 \( 1 + (15.0 + 8.70i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (13.5 + 23.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (42.2 - 24.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 9.84iT - 2.80e3T^{2} \)
59 \( 1 + (-76.1 - 43.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (29.9 + 51.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-3.85 + 6.68i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 79.5iT - 5.04e3T^{2} \)
73 \( 1 + 6.01T + 5.32e3T^{2} \)
79 \( 1 + (-8.46 - 14.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (127. - 73.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 100. iT - 7.92e3T^{2} \)
97 \( 1 + (36.3 + 62.8i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.411113871340089214827599642049, −8.712071595014055558751325821207, −8.229666612365280658823553527590, −7.15412806135949887857124842862, −6.44642788652642000221900622899, −5.36855822693016674614252996225, −4.71069475823371341386257382471, −3.74097756670995288966666129287, −2.53379285302098297199659520325, −1.55870713860363184469751995036, 0.10245361276552110268156499286, 1.40602867187980870502182329064, 2.72842114687080896247646824490, 3.78837620620937319987374464234, 4.56702667061155974081608325902, 5.42271336337949191431951436656, 6.65057721191797439175881754555, 7.15431316261144112297626508850, 8.103142642748293248794556898206, 8.579790682913764387829104809942

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