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Properties

Label 2-270-5.2-c2-0-7
Degree $2$
Conductor $270$
Sign $0.0640 - 0.997i$
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 + (4.07 + 2.89i)5-s + (1.68 + 1.68i)7-s + (−2 + 2i)8-s + (1.18 + 6.97i)10-s + 6.57·11-s + (−3.12 + 3.12i)13-s + 3.37i·14-s − 4·16-s + (−4.54 − 4.54i)17-s + 27.6i·19-s + (−5.79 + 8.15i)20-s + (6.57 + 6.57i)22-s + (13.7 − 13.7i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.815 + 0.579i)5-s + (0.241 + 0.241i)7-s + (−0.250 + 0.250i)8-s + (0.118 + 0.697i)10-s + 0.597·11-s + (−0.240 + 0.240i)13-s + 0.241i·14-s − 0.250·16-s + (−0.267 − 0.267i)17-s + 1.45i·19-s + (−0.289 + 0.407i)20-s + (0.298 + 0.298i)22-s + (0.598 − 0.598i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0640 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0640 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.0640 - 0.997i$
Analytic conductor: \(7.35696\)
Root analytic conductor: \(2.71237\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1),\ 0.0640 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73149 + 1.62398i\)
\(L(\frac12)\) \(\approx\) \(1.73149 + 1.62398i\)
\(L(2)\) 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 \( 1 + (-4.07 - 2.89i)T \)
good7 \( 1 + (-1.68 - 1.68i)T + 49iT^{2} \)
11 \( 1 - 6.57T + 121T^{2} \)
13 \( 1 + (3.12 - 3.12i)T - 169iT^{2} \)
17 \( 1 + (4.54 + 4.54i)T + 289iT^{2} \)
19 \( 1 - 27.6iT - 361T^{2} \)
23 \( 1 + (-13.7 + 13.7i)T - 529iT^{2} \)
29 \( 1 + 10.9iT - 841T^{2} \)
31 \( 1 + 36.4T + 961T^{2} \)
37 \( 1 + (-43.7 - 43.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 44.3T + 1.68e3T^{2} \)
43 \( 1 + (-2.47 + 2.47i)T - 1.84e3iT^{2} \)
47 \( 1 + (62.4 + 62.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (-29.0 + 29.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 41.6iT - 3.48e3T^{2} \)
61 \( 1 + 42.5T + 3.72e3T^{2} \)
67 \( 1 + (67.5 + 67.5i)T + 4.48e3iT^{2} \)
71 \( 1 - 47.6T + 5.04e3T^{2} \)
73 \( 1 + (-74.6 + 74.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 78.3iT - 6.24e3T^{2} \)
83 \( 1 + (-60.7 + 60.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 165. iT - 7.92e3T^{2} \)
97 \( 1 + (86.6 + 86.6i)T + 9.40e3iT^{2} \)
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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

−12.00794468341653283773841403926, −11.08888873901943947549532048990, −9.963265041699846009162459678723, −9.058592898015461302344271532801, −7.87980946331897309372401293452, −6.74605802370465633351659993238, −5.99893421934297209318266331829, −4.88492494782244520693807453934, −3.48038736584617824823912352944, −2.01690984077805322143531039911, 1.15590725822500198273414872454, 2.59915183235057420001729411581, 4.20761931143158167274027438257, 5.18633436167263272505344243061, 6.21983241766879191200862818854, 7.44202603578085256840308415098, 9.007541719730084408540429519737, 9.467329767062754079756331186491, 10.74905171824715883606056805481, 11.39213193302233657209898666090

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