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Properties

Label 2-270-5.3-c2-0-8
Degree $2$
Conductor $270$
Sign $0.608 + 0.793i$
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 + (1.77 − 4.67i)5-s + (−5.89 + 5.89i)7-s + (2 + 2i)8-s + (2.89 + 6.44i)10-s + 14.8·11-s + (−11.4 − 11.4i)13-s − 11.7i·14-s − 4·16-s + (14.2 − 14.2i)17-s − 10.5i·19-s + (−9.34 − 3.55i)20-s + (−14.8 + 14.8i)22-s + (−21.7 − 21.7i)23-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s − 0.5i·4-s + (0.355 − 0.934i)5-s + (−0.842 + 0.842i)7-s + (0.250 + 0.250i)8-s + (0.289 + 0.644i)10-s + 1.35·11-s + (−0.880 − 0.880i)13-s − 0.842i·14-s − 0.250·16-s + (0.836 − 0.836i)17-s − 0.555i·19-s + (−0.467 − 0.177i)20-s + (−0.677 + 0.677i)22-s + (−0.944 − 0.944i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.608 + 0.793i$
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: \((2,\ 270,\ (\ :1),\ 0.608 + 0.793i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.966146 - 0.476598i\)
\(L(\frac12)\) \(\approx\) \(0.966146 - 0.476598i\)
\(L(2)\) 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 \( 1 + (-1.77 + 4.67i)T \)
good7 \( 1 + (5.89 - 5.89i)T - 49iT^{2} \)
11 \( 1 - 14.8T + 121T^{2} \)
13 \( 1 + (11.4 + 11.4i)T + 169iT^{2} \)
17 \( 1 + (-14.2 + 14.2i)T - 289iT^{2} \)
19 \( 1 + 10.5iT - 361T^{2} \)
23 \( 1 + (21.7 + 21.7i)T + 529iT^{2} \)
29 \( 1 + 30.9iT - 841T^{2} \)
31 \( 1 - 57.9T + 961T^{2} \)
37 \( 1 + (-12.3 + 12.3i)T - 1.36e3iT^{2} \)
41 \( 1 - 10.4T + 1.68e3T^{2} \)
43 \( 1 + (2.94 + 2.94i)T + 1.84e3iT^{2} \)
47 \( 1 + (-26.6 + 26.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (10.6 + 10.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 60.0iT - 3.48e3T^{2} \)
61 \( 1 + 102.T + 3.72e3T^{2} \)
67 \( 1 + (46.4 - 46.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 38.9T + 5.04e3T^{2} \)
73 \( 1 + (-71.7 - 71.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 118. iT - 6.24e3T^{2} \)
83 \( 1 + (18.7 + 18.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 44.6iT - 7.92e3T^{2} \)
97 \( 1 + (125. - 125. i)T - 9.40e3iT^{2} \)
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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

−11.93593012385418128497699124139, −10.11342998705579410194135217788, −9.547785572222265675414770984967, −8.805817035911977729060851162457, −7.79280799727386889035520298386, −6.45018758105359997646088288977, −5.70185172667274138644636516150, −4.50133819529369572121659894453, −2.59627692801206798751038038475, −0.68039803764449209045751497143, 1.56723133109216685241954664015, 3.21368157856631765956364092682, 4.12078519184904608903829087674, 6.19137266775359397731608473071, 6.89371515378821989911168096215, 7.898941120789014393591245784266, 9.455478282454832904043519852825, 9.855939860845948894583335431491, 10.71537147119038759470994638824, 11.79717343485395037315648723342

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