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Properties

Label 2-270-5.3-c2-0-12
Degree $2$
Conductor $270$
Sign $0.245 + 0.969i$
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  + (−1 + i)2-s − 2i·4-s + (3.47 − 3.59i)5-s + (6.91 − 6.91i)7-s + (2 + 2i)8-s + (0.118 + 7.07i)10-s − 17.4·11-s + (−10.0 − 10.0i)13-s + 13.8i·14-s − 4·16-s + (−9.74 + 9.74i)17-s − 20.5i·19-s + (−7.18 − 6.95i)20-s + (17.4 − 17.4i)22-s + (0.673 + 0.673i)23-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 0.5i·4-s + (0.695 − 0.718i)5-s + (0.987 − 0.987i)7-s + (0.250 + 0.250i)8-s + (0.0118 + 0.707i)10-s − 1.58·11-s + (−0.774 − 0.774i)13-s + 0.987i·14-s − 0.250·16-s + (−0.573 + 0.573i)17-s − 1.07i·19-s + (−0.359 − 0.347i)20-s + (0.792 − 0.792i)22-s + (0.0292 + 0.0292i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(7.35696\)
Root analytic conductor: \(2.71237\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1),\ 0.245 + 0.969i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.913934 - 0.710978i\)
\(L(\frac12)\) \(\approx\) \(0.913934 - 0.710978i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 \)
5 \( 1 + (-3.47 + 3.59i)T \)
good7 \( 1 + (-6.91 + 6.91i)T - 49iT^{2} \)
11 \( 1 + 17.4T + 121T^{2} \)
13 \( 1 + (10.0 + 10.0i)T + 169iT^{2} \)
17 \( 1 + (9.74 - 9.74i)T - 289iT^{2} \)
19 \( 1 + 20.5iT - 361T^{2} \)
23 \( 1 + (-0.673 - 0.673i)T + 529iT^{2} \)
29 \( 1 + 16.6iT - 841T^{2} \)
31 \( 1 - 55.2T + 961T^{2} \)
37 \( 1 + (34.9 - 34.9i)T - 1.36e3iT^{2} \)
41 \( 1 - 54.1T + 1.68e3T^{2} \)
43 \( 1 + (3.08 + 3.08i)T + 1.84e3iT^{2} \)
47 \( 1 + (5.65 - 5.65i)T - 2.20e3iT^{2} \)
53 \( 1 + (45.4 + 45.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 94.1iT - 3.48e3T^{2} \)
61 \( 1 - 54.1T + 3.72e3T^{2} \)
67 \( 1 + (-33.3 + 33.3i)T - 4.48e3iT^{2} \)
71 \( 1 - 71.8T + 5.04e3T^{2} \)
73 \( 1 + (10.1 + 10.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 105. iT - 6.24e3T^{2} \)
83 \( 1 + (-86.4 - 86.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 12.4iT - 7.92e3T^{2} \)
97 \( 1 + (40.9 - 40.9i)T - 9.40e3iT^{2} \)
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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

−11.16910654564727674061402906936, −10.38905660546738026634482900131, −9.682596561215100468662802093073, −8.264418421227376663341561361376, −7.911543014078947000074152194644, −6.63058814484396951331323196951, −5.21163872928654980305866265414, −4.69657934892309397250991694702, −2.34437271974939728226819702891, −0.67161685326983419943729563936, 2.02281418403474007643836066484, 2.74639128598358044864235028200, 4.75543939025247428031016259734, 5.77841873597623537976064177470, 7.19573946465265657051786632460, 8.132900480789382347325073459927, 9.141672188804527126178231298312, 10.10985726684470504058584522381, 10.85548441629895627011224137279, 11.75491116225618316621920417347

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