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Properties

Label 2-270-45.32-c1-0-3
Degree $2$
Conductor $270$
Sign $0.958 - 0.285i$
Analytic cond. $2.15596$
Root an. cond. $1.46831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (1.51 − 1.64i)5-s + (−1.00 + 3.75i)7-s + (0.707 + 0.707i)8-s + (1.89 − 1.19i)10-s + (3.44 − 1.98i)11-s + (−0.256 − 0.956i)13-s + (−1.94 + 3.36i)14-s + (0.500 + 0.866i)16-s + (−0.120 + 0.120i)17-s − 1.88i·19-s + (2.13 − 0.661i)20-s + (3.83 − 1.02i)22-s + (−5.08 + 1.36i)23-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.249i)4-s + (0.679 − 0.733i)5-s + (−0.380 + 1.41i)7-s + (0.249 + 0.249i)8-s + (0.598 − 0.376i)10-s + (1.03 − 0.599i)11-s + (−0.0710 − 0.265i)13-s + (−0.519 + 0.899i)14-s + (0.125 + 0.216i)16-s + (−0.0291 + 0.0291i)17-s − 0.432i·19-s + (0.477 − 0.147i)20-s + (0.818 − 0.219i)22-s + (−1.06 + 0.284i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.958 - 0.285i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.958 - 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98231 + 0.288463i\)
\(L(\frac12)\) \(\approx\) \(1.98231 + 0.288463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-1.51 + 1.64i)T \)
good7 \( 1 + (1.00 - 3.75i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.44 + 1.98i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.256 + 0.956i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.120 - 0.120i)T - 17iT^{2} \)
19 \( 1 + 1.88iT - 19T^{2} \)
23 \( 1 + (5.08 - 1.36i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.15 + 3.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.70 - 8.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.26 - 3.26i)T + 37iT^{2} \)
41 \( 1 + (7.15 + 4.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.99 + 0.533i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.34 + 0.897i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.66 + 3.66i)T + 53iT^{2} \)
59 \( 1 + (-2.72 + 4.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.35 + 7.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.86 + 2.10i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.94iT - 71T^{2} \)
73 \( 1 + (8.27 - 8.27i)T - 73iT^{2} \)
79 \( 1 + (-11.7 + 6.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.81 - 6.75i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 4.87T + 89T^{2} \)
97 \( 1 + (0.387 - 1.44i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.14857973816520085340778100205, −11.40174894959616324195145876356, −9.902614279647338868822877458635, −9.025290754806859995246123037424, −8.286895839448296275814258760834, −6.59265288667186788220378930223, −5.83813923492515267397866075705, −5.00771264731238961890176127756, −3.47003054507189393889113433626, −1.99671374505171146934510913853, 1.81045652252277510644023774591, 3.49868943646231788586660239270, 4.36577484572442149481350930151, 5.97374943989535263547938490466, 6.78722631533523613523249547846, 7.57045359068021846399564111581, 9.428462444281366674643926692600, 10.10379950377345552245516560000, 10.92698989684610279959340844441, 11.87743113110075863273169460272

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