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Properties

Label 2-1620-45.29-c2-0-43
Degree $2$
Conductor $1620$
Sign $-0.772 + 0.634i$
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  + (2.15 − 4.50i)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 + (−15.6 − 19.4i)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  + (0.431 − 0.901i)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 + (−0.627 − 0.778i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.772 + 0.634i$
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: \((2,\ 1620,\ (\ :1),\ -0.772 + 0.634i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.093650828\)
\(L(\frac12)\) \(\approx\) \(1.093650828\)
\(L(2)\) 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 \( 1 + (-2.15 + 4.50i)T \)
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

−8.755588008657878780805367537648, −8.266511953710651642743127839904, −7.49107762827316099847753257344, −6.31556893571721617865616037116, −5.45001653759455025519917237210, −4.88585782735162305074198117621, −4.01767808209687718163385213331, −2.43257564906554391991370656006, −1.77643750143900699359707892434, −0.27039389894839776789269038739, 1.49404778553397154177596168266, 2.49746252958669663641170649248, 3.47233519216601494190220360159, 4.65201257582010022821926350965, 5.37796908951010597331881788427, 6.31636371599364670568968124617, 7.33524862645003344368152388037, 7.71209820914127259901959412709, 8.597058819407101519349870980295, 9.863582925586953977635443592089

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