Properties

Label 2-270-9.2-c10-0-2
Degree $2$
Conductor $270$
Sign $-0.974 - 0.224i$
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 + (7.82e3 − 1.35e4i)7-s + 1.15e4i·8-s + 3.16e4·10-s + (−1.94e5 − 1.12e5i)11-s + (−8.64e4 − 1.49e5i)13-s + (3.06e5 − 1.77e5i)14-s + (−1.31e5 + 2.27e5i)16-s − 6.84e5i·17-s + 1.00e6·19-s + (6.19e5 + 3.57e5i)20-s + (−2.53e6 − 4.39e6i)22-s + (−1.70e6 + 9.87e5i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (0.465 − 0.806i)7-s + 0.353i·8-s + 0.316·10-s + (−1.20 − 0.696i)11-s + (−0.232 − 0.403i)13-s + (0.570 − 0.329i)14-s + (−0.125 + 0.216i)16-s − 0.482i·17-s + 0.404·19-s + (0.193 + 0.111i)20-s + (−0.492 − 0.853i)22-s + (−0.265 + 0.153i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $-0.974 - 0.224i$
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: \((2,\ 270,\ (\ :5),\ -0.974 - 0.224i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.2353119817\)
\(L(\frac12)\) \(\approx\) \(0.2353119817\)
\(L(6)\) 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 \( 1 + (-7.82e3 + 1.35e4i)T + (-1.41e8 - 2.44e8i)T^{2} \)
11 \( 1 + (1.94e5 + 1.12e5i)T + (1.29e10 + 2.24e10i)T^{2} \)
13 \( 1 + (8.64e4 + 1.49e5i)T + (-6.89e10 + 1.19e11i)T^{2} \)
17 \( 1 + 6.84e5iT - 2.01e12T^{2} \)
19 \( 1 - 1.00e6T + 6.13e12T^{2} \)
23 \( 1 + (1.70e6 - 9.87e5i)T + (2.07e13 - 3.58e13i)T^{2} \)
29 \( 1 + (1.06e7 + 6.12e6i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (-8.29e6 - 1.43e7i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 + 7.30e7T + 4.80e15T^{2} \)
41 \( 1 + (1.57e8 - 9.07e7i)T + (6.71e15 - 1.16e16i)T^{2} \)
43 \( 1 + (7.19e7 - 1.24e8i)T + (-1.08e16 - 1.87e16i)T^{2} \)
47 \( 1 + (6.39e7 + 3.69e7i)T + (2.62e16 + 4.55e16i)T^{2} \)
53 \( 1 - 1.54e8iT - 1.74e17T^{2} \)
59 \( 1 + (3.56e8 - 2.06e8i)T + (2.55e17 - 4.42e17i)T^{2} \)
61 \( 1 + (-5.82e8 + 1.00e9i)T + (-3.56e17 - 6.17e17i)T^{2} \)
67 \( 1 + (-9.22e8 - 1.59e9i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 - 3.30e9iT - 3.25e18T^{2} \)
73 \( 1 - 6.65e8T + 4.29e18T^{2} \)
79 \( 1 + (-2.00e9 + 3.47e9i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (-4.94e9 - 2.85e9i)T + (7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 - 9.92e9iT - 3.11e19T^{2} \)
97 \( 1 + (3.64e9 - 6.31e9i)T + (-3.68e19 - 6.38e19i)T^{2} \)
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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

−10.66050793444652878593196291971, −9.787155931141096923784314486351, −8.374984506227237943427957118038, −7.71442127035821927674129938488, −6.68441532337564347309833918546, −5.44250361766568605178862140989, −4.90336481151108579589699196822, −3.59179170554665490957275437227, −2.57093470960159414697876498838, −1.20475716959833430721946610455, 0.03164599669777026033657151269, 1.78883605734254776996431932321, 2.30886271632313435918387573303, 3.52886676131880305381077344394, 4.92094102670814773863849307098, 5.45548208492852912977405635246, 6.63910447145840089339982091908, 7.73305713624425952686241138549, 8.860374637864506126507209050310, 9.971593823419459515396319194718

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