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Properties

Label 2-1620-45.29-c2-0-37
Degree $2$
Conductor $1620$
Sign $0.944 - 0.329i$
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  + (4.29 + 2.55i)5-s + (2.42 + 1.39i)7-s + (15.7 + 9.07i)11-s + (19.9 − 11.5i)13-s + 5.72·17-s + 23.1·19-s + (−0.135 − 0.235i)23-s + (11.8 + 21.9i)25-s + (−34.4 − 19.8i)29-s + (−23.6 − 41.0i)31-s + (6.82 + 12.2i)35-s + 34.8i·37-s + (11.4 − 6.63i)41-s + (−40.4 − 23.3i)43-s + (20.4 − 35.4i)47-s + ⋯
L(s)  = 1  + (0.859 + 0.511i)5-s + (0.345 + 0.199i)7-s + (1.42 + 0.824i)11-s + (1.53 − 0.885i)13-s + 0.336·17-s + 1.22·19-s + (−0.00590 − 0.0102i)23-s + (0.475 + 0.879i)25-s + (−1.18 − 0.686i)29-s + (−0.764 − 1.32i)31-s + (0.194 + 0.348i)35-s + 0.942i·37-s + (0.280 − 0.161i)41-s + (−0.940 − 0.543i)43-s + (0.435 − 0.753i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.944 - 0.329i$
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: \((2,\ 1620,\ (\ :1),\ 0.944 - 0.329i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.177441014\)
\(L(\frac12)\) \(\approx\) \(3.177441014\)
\(L(2)\) 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 \( 1 + (-4.29 - 2.55i)T \)
good7 \( 1 + (-2.42 - 1.39i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-15.7 - 9.07i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-19.9 + 11.5i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 5.72T + 289T^{2} \)
19 \( 1 - 23.1T + 361T^{2} \)
23 \( 1 + (0.135 + 0.235i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (34.4 + 19.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (23.6 + 41.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 34.8iT - 1.36e3T^{2} \)
41 \( 1 + (-11.4 + 6.63i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (40.4 + 23.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-20.4 + 35.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 91.3T + 2.80e3T^{2} \)
59 \( 1 + (68.2 - 39.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (15.5 - 27.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-5.98 + 3.45i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 81.5iT - 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 + (31.7 - 55.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-0.142 + 0.246i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 28.5iT - 7.92e3T^{2} \)
97 \( 1 + (80.4 + 46.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.367334873914485242145727851346, −8.568503305755766185425197134232, −7.56446972812239388573352604528, −6.81912311854381671393788141940, −5.88761395469567142520622839393, −5.44214594467148737296654091039, −4.05281412715914100457265467929, −3.30304529723378554087467008721, −1.99620602658490960523931187767, −1.14658355645684011104024560878, 1.17103388516827104620984622385, 1.58610014275743835822756749359, 3.30693388699230207718030736038, 4.02025551873736863074309268092, 5.16826349158760755581360961997, 5.94834910424480215170101995129, 6.57928266142150324745194937111, 7.56030638136181076688100630432, 8.797002368713512513601041255413, 8.958344734456725123345274208373

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