Properties

Label 2-270-9.2-c10-0-33
Degree $2$
Conductor $270$
Sign $0.214 + 0.976i$
Analytic cond. $171.546$
Root an. cond. $13.0975$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (19.5 + 11.3i)2-s + (255. + 443. i)4-s + (1.21e3 − 698. i)5-s + (−1.22e4 + 2.12e4i)7-s + 1.15e4i·8-s + 3.16e4·10-s + (−8.70e4 − 5.02e4i)11-s + (2.48e5 + 4.30e5i)13-s + (−4.81e5 + 2.78e5i)14-s + (−1.31e5 + 2.27e5i)16-s − 1.31e6i·17-s − 3.30e6·19-s + (6.19e5 + 3.57e5i)20-s + (−1.13e6 − 1.97e6i)22-s + (−7.41e5 + 4.27e5i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.731 + 1.26i)7-s + 0.353i·8-s + 0.316·10-s + (−0.540 − 0.312i)11-s + (0.670 + 1.16i)13-s + (−0.896 + 0.517i)14-s + (−0.125 + 0.216i)16-s − 0.928i·17-s − 1.33·19-s + (0.193 + 0.111i)20-s + (−0.220 − 0.382i)22-s + (−0.115 + 0.0664i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.214 + 0.976i$
Analytic conductor: \(171.546\)
Root analytic conductor: \(13.0975\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :5),\ 0.214 + 0.976i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.7916707733\)
\(L(\frac12)\) \(\approx\) \(0.7916707733\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-19.5 - 11.3i)T \)
3 \( 1 \)
5 \( 1 + (-1.21e3 + 698. i)T \)
good7 \( 1 + (1.22e4 - 2.12e4i)T + (-1.41e8 - 2.44e8i)T^{2} \)
11 \( 1 + (8.70e4 + 5.02e4i)T + (1.29e10 + 2.24e10i)T^{2} \)
13 \( 1 + (-2.48e5 - 4.30e5i)T + (-6.89e10 + 1.19e11i)T^{2} \)
17 \( 1 + 1.31e6iT - 2.01e12T^{2} \)
19 \( 1 + 3.30e6T + 6.13e12T^{2} \)
23 \( 1 + (7.41e5 - 4.27e5i)T + (2.07e13 - 3.58e13i)T^{2} \)
29 \( 1 + (2.44e7 + 1.40e7i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (3.92e6 + 6.79e6i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 - 8.06e7T + 4.80e15T^{2} \)
41 \( 1 + (4.04e7 - 2.33e7i)T + (6.71e15 - 1.16e16i)T^{2} \)
43 \( 1 + (9.45e7 - 1.63e8i)T + (-1.08e16 - 1.87e16i)T^{2} \)
47 \( 1 + (1.68e7 + 9.72e6i)T + (2.62e16 + 4.55e16i)T^{2} \)
53 \( 1 + 3.26e8iT - 1.74e17T^{2} \)
59 \( 1 + (-8.59e8 + 4.96e8i)T + (2.55e17 - 4.42e17i)T^{2} \)
61 \( 1 + (-2.34e8 + 4.05e8i)T + (-3.56e17 - 6.17e17i)T^{2} \)
67 \( 1 + (-1.23e9 - 2.13e9i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + 1.82e9iT - 3.25e18T^{2} \)
73 \( 1 + 1.52e9T + 4.29e18T^{2} \)
79 \( 1 + (-2.30e9 + 3.98e9i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (-1.55e9 - 8.98e8i)T + (7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + 4.51e9iT - 3.11e19T^{2} \)
97 \( 1 + (6.08e9 - 1.05e10i)T + (-3.68e19 - 6.38e19i)T^{2} \)
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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

−9.720110949278091264417371040072, −8.999360932079496019874355893655, −8.093462361444300236941025279371, −6.65065083924614574372219536825, −6.04810858388679888373712627992, −5.14153724814326952153159771380, −3.97242822516308045004981407192, −2.73056896943223767580494401980, −1.93576033659364254309948926972, −0.11950106311498524180827092339, 0.993555261508021631750045798040, 2.20882579440834353817896974638, 3.41719677670362291132686238503, 4.09998501550827080161076267147, 5.45606032538402124923276798771, 6.35551413634460979630028799211, 7.26804284398041393162573737608, 8.423465063317756300208683611245, 9.837404217724068994192566173084, 10.55496013314790490337818334971

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