Properties

Label 2-270-9.2-c10-0-35
Degree $2$
Conductor $270$
Sign $0.374 + 0.927i$
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 + (1.55e4 − 2.69e4i)7-s + 1.15e4i·8-s − 3.16e4·10-s + (−2.34e5 − 1.35e5i)11-s + (2.76e5 + 4.79e5i)13-s + (6.09e5 − 3.51e5i)14-s + (−1.31e5 + 2.27e5i)16-s + 5.34e5i·17-s + 1.67e5·19-s + (−6.19e5 − 3.57e5i)20-s + (−3.06e6 − 5.31e6i)22-s + (7.47e5 − 4.31e5i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.924 − 1.60i)7-s + 0.353i·8-s − 0.316·10-s + (−1.45 − 0.841i)11-s + (0.746 + 1.29i)13-s + (1.13 − 0.654i)14-s + (−0.125 + 0.216i)16-s + 0.376i·17-s + 0.0676·19-s + (−0.193 − 0.111i)20-s + (−0.595 − 1.03i)22-s + (0.116 − 0.0670i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.374 + 0.927i$
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: \((2,\ 270,\ (\ :5),\ 0.374 + 0.927i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.513348263\)
\(L(\frac12)\) \(\approx\) \(2.513348263\)
\(L(6)\) 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 \( 1 + (-1.55e4 + 2.69e4i)T + (-1.41e8 - 2.44e8i)T^{2} \)
11 \( 1 + (2.34e5 + 1.35e5i)T + (1.29e10 + 2.24e10i)T^{2} \)
13 \( 1 + (-2.76e5 - 4.79e5i)T + (-6.89e10 + 1.19e11i)T^{2} \)
17 \( 1 - 5.34e5iT - 2.01e12T^{2} \)
19 \( 1 - 1.67e5T + 6.13e12T^{2} \)
23 \( 1 + (-7.47e5 + 4.31e5i)T + (2.07e13 - 3.58e13i)T^{2} \)
29 \( 1 + (-2.04e7 - 1.17e7i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (2.24e6 + 3.89e6i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 - 5.34e7T + 4.80e15T^{2} \)
41 \( 1 + (-5.45e7 + 3.15e7i)T + (6.71e15 - 1.16e16i)T^{2} \)
43 \( 1 + (-2.64e7 + 4.58e7i)T + (-1.08e16 - 1.87e16i)T^{2} \)
47 \( 1 + (-3.24e7 - 1.87e7i)T + (2.62e16 + 4.55e16i)T^{2} \)
53 \( 1 + 7.23e8iT - 1.74e17T^{2} \)
59 \( 1 + (9.77e8 - 5.64e8i)T + (2.55e17 - 4.42e17i)T^{2} \)
61 \( 1 + (6.92e8 - 1.19e9i)T + (-3.56e17 - 6.17e17i)T^{2} \)
67 \( 1 + (5.79e8 + 1.00e9i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + 9.14e8iT - 3.25e18T^{2} \)
73 \( 1 - 4.31e8T + 4.29e18T^{2} \)
79 \( 1 + (-2.08e9 + 3.61e9i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (5.62e9 + 3.24e9i)T + (7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + 4.48e9iT - 3.11e19T^{2} \)
97 \( 1 + (-4.77e9 + 8.26e9i)T + (-3.68e19 - 6.38e19i)T^{2} \)
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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

−10.41595179442815096918147893032, −8.674708833753204729516397497162, −7.84778761677817508831673248251, −7.15461147942135358603390030083, −6.07594933856010969025160910540, −4.78650885319327852692028882586, −4.10889205748371662908941936332, −3.07952029414085731136784575184, −1.59784924799229735248195750806, −0.39537734584099128040652318154, 1.05754941074873158775402280000, 2.34717268151315611394837399950, 2.95675790640286191405845436406, 4.58214281836201850918297097986, 5.24747203131177979274625797342, 6.01683261342983499651888127673, 7.72240324742271805612126928569, 8.259674704965521507723789596469, 9.448234473624024920874878691418, 10.61398781697057775726374949844

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