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Properties

Label 2-270-45.43-c2-0-2
Degree $2$
Conductor $270$
Sign $-0.747 - 0.664i$
Analytic cond. $7.35696$
Root an. cond. $2.71237$
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  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (−1.44 − 4.78i)5-s + (1.61 + 6.01i)7-s + (−2 − 1.99i)8-s + (6.00 − 3.73i)10-s + (−5.39 + 9.34i)11-s + (−6.33 + 23.6i)13-s + (−7.63 + 4.40i)14-s + (1.99 − 3.46i)16-s + (11.1 − 11.1i)17-s + 20.2i·19-s + (7.29 + 6.83i)20-s + (−14.7 − 3.94i)22-s + (−7.01 + 26.1i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.250i)4-s + (−0.289 − 0.957i)5-s + (0.230 + 0.859i)7-s + (−0.250 − 0.249i)8-s + (0.600 − 0.373i)10-s + (−0.490 + 0.849i)11-s + (−0.487 + 1.81i)13-s + (−0.545 + 0.314i)14-s + (0.124 − 0.216i)16-s + (0.656 − 0.656i)17-s + 1.06i·19-s + (0.364 + 0.341i)20-s + (−0.669 − 0.179i)22-s + (−0.305 + 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $-0.747 - 0.664i$
Analytic conductor: \(7.35696\)
Root analytic conductor: \(2.71237\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1),\ -0.747 - 0.664i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.413343 + 1.08683i\)
\(L(\frac12)\) \(\approx\) \(0.413343 + 1.08683i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 \)
5 \( 1 + (1.44 + 4.78i)T \)
good7 \( 1 + (-1.61 - 6.01i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (5.39 - 9.34i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (6.33 - 23.6i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (-11.1 + 11.1i)T - 289iT^{2} \)
19 \( 1 - 20.2iT - 361T^{2} \)
23 \( 1 + (7.01 - 26.1i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (17.5 + 10.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (5.98 + 10.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-40.1 + 40.1i)T - 1.36e3iT^{2} \)
41 \( 1 + (-5.77 - 10.0i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (18.9 - 5.07i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-4.89 - 18.2i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (31.5 + 31.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (49.2 - 28.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (15.9 - 27.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-79.8 - 21.3i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 22.4T + 5.04e3T^{2} \)
73 \( 1 + (-11.7 - 11.7i)T + 5.32e3iT^{2} \)
79 \( 1 + (-31.0 - 17.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-115. + 30.9i)T + (5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 12.2iT - 7.92e3T^{2} \)
97 \( 1 + (21.4 + 80.1i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−12.10194663290119826769793118504, −11.55071238317207125851411281232, −9.620961689204282345416313918025, −9.280523934718833491406007439873, −8.025737549422648949701405484786, −7.33006333750385457696433177436, −5.87189455446492316598227274974, −4.99846624902091624430971444740, −4.02212183553168022052034100610, −1.94902672512416685216184452865, 0.55368832144223899003086666249, 2.72954624807512661199148186853, 3.58256881649687878036469093883, 5.00523345703243837724520209654, 6.23926068056757842523793235302, 7.58058769544714976393541803739, 8.269590628655325899073361737243, 9.867278829720148609462592136878, 10.74665700473149163727435393732, 10.91478541607051162053823335177

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