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Properties

Label 2-270-45.43-c2-0-6
Degree $2$
Conductor $270$
Sign $0.0668 + 0.997i$
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  + (−0.366 − 1.36i)2-s + (−1.73 + i)4-s + (−3.85 + 3.18i)5-s + (0.348 + 1.30i)7-s + (2 + 1.99i)8-s + (5.76 + 4.10i)10-s + (7.62 − 13.2i)11-s + (2.13 − 7.95i)13-s + (1.64 − 0.952i)14-s + (1.99 − 3.46i)16-s + (20.4 − 20.4i)17-s − 3.08i·19-s + (3.49 − 9.37i)20-s + (−20.8 − 5.58i)22-s + (0.369 − 1.37i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.250i)4-s + (−0.770 + 0.636i)5-s + (0.0497 + 0.185i)7-s + (0.250 + 0.249i)8-s + (0.576 + 0.410i)10-s + (0.693 − 1.20i)11-s + (0.163 − 0.611i)13-s + (0.117 − 0.0680i)14-s + (0.124 − 0.216i)16-s + (1.20 − 1.20i)17-s − 0.162i·19-s + (0.174 − 0.468i)20-s + (−0.947 − 0.253i)22-s + (0.0160 − 0.0599i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0668 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0668 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.0668 + 0.997i$
Analytic conductor: \(7.35696\)
Root analytic conductor: \(2.71237\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1),\ 0.0668 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.855650 - 0.800213i\)
\(L(\frac12)\) \(\approx\) \(0.855650 - 0.800213i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
3 \( 1 \)
5 \( 1 + (3.85 - 3.18i)T \)
good7 \( 1 + (-0.348 - 1.30i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-7.62 + 13.2i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-2.13 + 7.95i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (-20.4 + 20.4i)T - 289iT^{2} \)
19 \( 1 + 3.08iT - 361T^{2} \)
23 \( 1 + (-0.369 + 1.37i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (-24.9 - 14.3i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (8.90 + 15.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (9.95 - 9.95i)T - 1.36e3iT^{2} \)
41 \( 1 + (-33.0 - 57.3i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (5.43 - 1.45i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (21.3 + 79.8i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (68.0 + 68.0i)T + 2.80e3iT^{2} \)
59 \( 1 + (-24.5 + 14.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (19.4 - 33.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-119. - 31.9i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 66.5T + 5.04e3T^{2} \)
73 \( 1 + (61.7 + 61.7i)T + 5.32e3iT^{2} \)
79 \( 1 + (-48.5 - 28.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-98.9 + 26.5i)T + (5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 66.2iT - 7.92e3T^{2} \)
97 \( 1 + (7.64 + 28.5i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−11.55581395897646621921856184785, −10.68448846756448261069582599170, −9.715427392409846811902548551804, −8.574726304473016238872969658094, −7.78324954527006100178855481511, −6.56054717682553315664119412698, −5.19680688243976299470092308825, −3.66679406522618402385405332061, −2.90047951480728396756254344247, −0.74569853125259766037318095976, 1.37138477799522121915614445997, 3.84389362001451771838040415754, 4.65657764479674060000134790901, 5.99597302881972360787104293543, 7.17786099434582384602296411829, 7.943513340342666315029910844081, 8.934523224709364554109818842629, 9.797033097556457376371580478161, 10.92963594740794142505078082256, 12.27699772822821438942445609456

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