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Properties

Label 2-1620-45.29-c2-0-14
Degree $2$
Conductor $1620$
Sign $-0.0622 - 0.998i$
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  + (1.17 + 4.86i)5-s + (−5.87 − 3.39i)7-s + (−8.57 − 4.94i)11-s + (17.6 − 10.1i)13-s − 19.1·17-s + 12·19-s + (4.79 + 8.30i)23-s + (−22.2 + 11.4i)25-s + (7.34 + 4.24i)29-s + (19 + 32.9i)31-s + (9.59 − 32.5i)35-s + 6.78i·37-s + (60.0 − 34.6i)41-s + (58.7 + 33.9i)43-s + (−38.3 + 66.4i)47-s + ⋯
L(s)  = 1  + (0.234 + 0.972i)5-s + (−0.839 − 0.484i)7-s + (−0.779 − 0.449i)11-s + (1.35 − 0.782i)13-s − 1.12·17-s + 0.631·19-s + (0.208 + 0.361i)23-s + (−0.889 + 0.456i)25-s + (0.253 + 0.146i)29-s + (0.612 + 1.06i)31-s + (0.274 − 0.929i)35-s + 0.183i·37-s + (1.46 − 0.845i)41-s + (1.36 + 0.788i)43-s + (−0.816 + 1.41i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0622 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0622 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.0622 - 0.998i$
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: \((2,\ 1620,\ (\ :1),\ -0.0622 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.320713328\)
\(L(\frac12)\) \(\approx\) \(1.320713328\)
\(L(2)\) 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 \( 1 + (-1.17 - 4.86i)T \)
good7 \( 1 + (5.87 + 3.39i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.57 + 4.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-17.6 + 10.1i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 19.1T + 289T^{2} \)
19 \( 1 - 12T + 361T^{2} \)
23 \( 1 + (-4.79 - 8.30i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-7.34 - 4.24i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-19 - 32.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 6.78iT - 1.36e3T^{2} \)
41 \( 1 + (-60.0 + 34.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-58.7 - 33.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (38.3 - 66.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + (72.2 - 41.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-35 + 60.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (93.9 - 54.2i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 118. iT - 5.04e3T^{2} \)
73 \( 1 - 13.5iT - 5.32e3T^{2} \)
79 \( 1 + (15 - 25.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (67.1 - 116. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 32.5iT - 7.92e3T^{2} \)
97 \( 1 + (82.2 + 47.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.522725204386853437285046247505, −8.613651365989347126642134764027, −7.74079145264461133136762136617, −6.95019982279000568362515999276, −6.19647790587572525254470951856, −5.59411178051035983212890308175, −4.23705950047815876210836580111, −3.22419862735086994775839847240, −2.72796628090727211821543204044, −1.06044373382658988661719329003, 0.40145754742198908905967184349, 1.78573601149878640828767913137, 2.81065831424944422154419004166, 4.08642332212503651185787652293, 4.77555059456880307561890446697, 5.90602093054533979718524369648, 6.32379744052375152485739358191, 7.46868995006450602007948645973, 8.388446311275450759821450870577, 9.105975942272752021272937741634

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