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Properties

Label 2-1620-45.29-c2-0-3
Degree $2$
Conductor $1620$
Sign $-0.871 - 0.491i$
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 + 0.385i)5-s + (−6.51 − 3.76i)7-s + (−5.77 − 3.33i)11-s + (13.4 − 7.78i)13-s − 18.9·17-s − 27.1·19-s + (9.80 + 16.9i)23-s + (24.7 + 3.84i)25-s + (6.03 + 3.48i)29-s + (−14.7 − 25.5i)31-s + (−31.0 − 21.2i)35-s + 52.2i·37-s + (−52.7 + 30.4i)41-s + (−9.23 − 5.33i)43-s + (−37.9 + 65.6i)47-s + ⋯
L(s)  = 1  + (0.997 + 0.0770i)5-s + (−0.931 − 0.537i)7-s + (−0.525 − 0.303i)11-s + (1.03 − 0.598i)13-s − 1.11·17-s − 1.42·19-s + (0.426 + 0.738i)23-s + (0.988 + 0.153i)25-s + (0.208 + 0.120i)29-s + (−0.476 − 0.824i)31-s + (−0.887 − 0.607i)35-s + 1.41i·37-s + (−1.28 + 0.742i)41-s + (−0.214 − 0.123i)43-s + (−0.806 + 1.39i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.871 - 0.491i$
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: \((2,\ 1620,\ (\ :1),\ -0.871 - 0.491i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2345157476\)
\(L(\frac12)\) \(\approx\) \(0.2345157476\)
\(L(2)\) 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 \( 1 + (-4.98 - 0.385i)T \)
good7 \( 1 + (6.51 + 3.76i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (5.77 + 3.33i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-13.4 + 7.78i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 18.9T + 289T^{2} \)
19 \( 1 + 27.1T + 361T^{2} \)
23 \( 1 + (-9.80 - 16.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-6.03 - 3.48i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.7 + 25.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 52.2iT - 1.36e3T^{2} \)
41 \( 1 + (52.7 - 30.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (9.23 + 5.33i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (37.9 - 65.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 47.3T + 2.80e3T^{2} \)
59 \( 1 + (-21.1 + 12.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (20.3 - 35.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (51.0 - 29.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 33.9iT - 5.04e3T^{2} \)
73 \( 1 - 79.8iT - 5.32e3T^{2} \)
79 \( 1 + (43.0 - 74.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-76.5 + 132. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 83.3iT - 7.92e3T^{2} \)
97 \( 1 + (14.0 + 8.10i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.587402404554483953313224265382, −8.790822834682854782677604413871, −8.098977097151893906951748091071, −6.83569387303634532121129098751, −6.38420868649161217335661793517, −5.63865818759983402900085311533, −4.57257392929355928510000920694, −3.47435096601508822662099262012, −2.62453373291706096123845430522, −1.38305465853008008945014027437, 0.05832383763287001598631125476, 1.83046803224117189671768593675, 2.53259107612037838921227544055, 3.73040063691087372565179477004, 4.80110760043261977832241799154, 5.71737267799406148791123068943, 6.55986322066320233626961990971, 6.84268384468344049580771491053, 8.476622835455672057767084689631, 8.856930100025742825186667030546

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