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Properties

Label 2-270-5.3-c2-0-14
Degree $2$
Conductor $270$
Sign $-0.899 + 0.437i$
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 + (−4.22 − 2.67i)5-s + (3.89 − 3.89i)7-s + (−2 − 2i)8-s + (−6.89 + 1.55i)10-s − 5.10·11-s + (−6.55 − 6.55i)13-s − 7.79i·14-s − 4·16-s + (−11.7 + 11.7i)17-s − 15.4i·19-s + (−5.34 + 8.44i)20-s + (−5.10 + 5.10i)22-s + (−29.7 − 29.7i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.5i·4-s + (−0.844 − 0.534i)5-s + (0.556 − 0.556i)7-s + (−0.250 − 0.250i)8-s + (−0.689 + 0.155i)10-s − 0.463·11-s + (−0.503 − 0.503i)13-s − 0.556i·14-s − 0.250·16-s + (−0.692 + 0.692i)17-s − 0.813i·19-s + (−0.267 + 0.422i)20-s + (−0.231 + 0.231i)22-s + (−1.29 − 1.29i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $-0.899 + 0.437i$
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: \((2,\ 270,\ (\ :1),\ -0.899 + 0.437i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.294078 - 1.27638i\)
\(L(\frac12)\) \(\approx\) \(0.294078 - 1.27638i\)
\(L(2)\) 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 \( 1 + (4.22 + 2.67i)T \)
good7 \( 1 + (-3.89 + 3.89i)T - 49iT^{2} \)
11 \( 1 + 5.10T + 121T^{2} \)
13 \( 1 + (6.55 + 6.55i)T + 169iT^{2} \)
17 \( 1 + (11.7 - 11.7i)T - 289iT^{2} \)
19 \( 1 + 15.4iT - 361T^{2} \)
23 \( 1 + (29.7 + 29.7i)T + 529iT^{2} \)
29 \( 1 + 22.9iT - 841T^{2} \)
31 \( 1 - 4.05T + 961T^{2} \)
37 \( 1 + (-41.6 + 41.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 38.4T + 1.68e3T^{2} \)
43 \( 1 + (-50.9 - 50.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (-2.69 + 2.69i)T - 2.20e3iT^{2} \)
53 \( 1 + (-3.32 - 3.32i)T + 2.80e3iT^{2} \)
59 \( 1 + 15.9iT - 3.48e3T^{2} \)
61 \( 1 + 23.8T + 3.72e3T^{2} \)
67 \( 1 + (-56.4 + 56.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 58.9T + 5.04e3T^{2} \)
73 \( 1 + (99.7 + 99.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 112. iT - 6.24e3T^{2} \)
83 \( 1 + (-65.2 - 65.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 103. iT - 7.92e3T^{2} \)
97 \( 1 + (-89.7 + 89.7i)T - 9.40e3iT^{2} \)
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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

−11.21188818238188307048344880298, −10.73115134231777495382848745816, −9.504135342273131122002365553594, −8.260384404375885526836948300848, −7.53686994919812592490486962824, −6.07109390725795144398890333702, −4.68062575477943602920989826305, −4.12115860984121300617571731787, −2.45382239043347928166646311904, −0.55128568426325968323462635050, 2.42282498129323219804092383031, 3.84382515968381058111163127771, 4.92798722646335652355918428467, 6.08151700244774083159618285032, 7.31235213933870854933765656221, 7.928476597696558788032195970949, 9.026469236922594213143667783563, 10.32296150098542518835609762587, 11.59291958856209071845690186638, 11.86600321564084113522510677782

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