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Properties

Label 2-270-45.7-c2-0-7
Degree $2$
Conductor $270$
Sign $0.912 + 0.408i$
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (4.80 − 1.38i)5-s + (11.2 − 3.02i)7-s + (−1.99 + 2i)8-s + (−6.05 + 3.65i)10-s + (1.29 − 2.24i)11-s + (−18.3 − 4.90i)13-s + (−14.3 + 8.26i)14-s + (1.99 − 3.46i)16-s + (6.98 + 6.98i)17-s − 15.8i·19-s + (6.93 − 7.20i)20-s + (−0.950 + 3.54i)22-s + (2.99 + 0.802i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.250i)4-s + (0.960 − 0.277i)5-s + (1.61 − 0.432i)7-s + (−0.249 + 0.250i)8-s + (−0.605 + 0.365i)10-s + (0.118 − 0.204i)11-s + (−1.40 − 0.377i)13-s + (−1.02 + 0.590i)14-s + (0.124 − 0.216i)16-s + (0.410 + 0.410i)17-s − 0.833i·19-s + (0.346 − 0.360i)20-s + (−0.0431 + 0.161i)22-s + (0.130 + 0.0349i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.912 + 0.408i$
Analytic conductor: \(7.35696\)
Root analytic conductor: \(2.71237\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1),\ 0.912 + 0.408i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.52678 - 0.325960i\)
\(L(\frac12)\) \(\approx\) \(1.52678 - 0.325960i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 \)
5 \( 1 + (-4.80 + 1.38i)T \)
good7 \( 1 + (-11.2 + 3.02i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.29 + 2.24i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (18.3 + 4.90i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (-6.98 - 6.98i)T + 289iT^{2} \)
19 \( 1 + 15.8iT - 361T^{2} \)
23 \( 1 + (-2.99 - 0.802i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + (-27.9 - 16.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (8.35 + 14.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-20.7 - 20.7i)T + 1.36e3iT^{2} \)
41 \( 1 + (30.6 + 53.0i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-16.7 - 62.4i)T + (-1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (-4.44 + 1.19i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-19.2 + 19.2i)T - 2.80e3iT^{2} \)
59 \( 1 + (2.53 - 1.46i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-45.1 + 78.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (31.9 - 119. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 96.4T + 5.04e3T^{2} \)
73 \( 1 + (-2.64 + 2.64i)T - 5.32e3iT^{2} \)
79 \( 1 + (-20.6 - 11.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-28.1 - 104. i)T + (-5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 51.4iT - 7.92e3T^{2} \)
97 \( 1 + (124. - 33.4i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.42434159845802994528801733685, −10.53274802853099032059707179030, −9.763971751097286697944083422302, −8.699083426867103294216964572719, −7.86291792575903577192041247223, −6.88032885196617607115379452180, −5.47410111724003400660868321660, −4.69264259461436477816903411957, −2.44367878276576065441566105331, −1.14994256659498109504369896035, 1.59052438784475340110010287434, 2.57489751960674457437057035364, 4.67511462419559167348767770550, 5.67618644498229001348386693054, 7.05402114130424852166885369121, 7.952146210904444248693727559199, 8.984711834630887404959757048252, 9.885682087361845379312426227819, 10.65123495142702996377534484317, 11.75673713791915381410814806144

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