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Properties

Label 2-270-135.122-c1-0-17
Degree $2$
Conductor $270$
Sign $0.0762 + 0.997i$
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  + (0.819 + 0.573i)2-s + (−0.738 − 1.56i)3-s + (0.342 + 0.939i)4-s + (−0.334 − 2.21i)5-s + (0.293 − 1.70i)6-s + (−1.37 − 2.94i)7-s + (−0.258 + 0.965i)8-s + (−1.90 + 2.31i)9-s + (0.994 − 2.00i)10-s + (0.631 + 0.752i)11-s + (1.21 − 1.22i)12-s + (−1.82 − 2.60i)13-s + (0.563 − 3.19i)14-s + (−3.21 + 2.15i)15-s + (−0.766 + 0.642i)16-s + (−0.889 − 3.31i)17-s + ⋯
L(s)  = 1  + (0.579 + 0.405i)2-s + (−0.426 − 0.904i)3-s + (0.171 + 0.469i)4-s + (−0.149 − 0.988i)5-s + (0.119 − 0.696i)6-s + (−0.518 − 1.11i)7-s + (−0.0915 + 0.341i)8-s + (−0.636 + 0.771i)9-s + (0.314 − 0.633i)10-s + (0.190 + 0.226i)11-s + (0.352 − 0.355i)12-s + (−0.505 − 0.722i)13-s + (0.150 − 0.854i)14-s + (−0.830 + 0.556i)15-s + (−0.191 + 0.160i)16-s + (−0.215 − 0.805i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0762 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0762 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270\)    =    \(2 \cdot 3^{3} \cdot 5\)
Sign: $0.0762 + 0.997i$
Analytic conductor: \(2.15596\)
Root analytic conductor: \(1.46831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{270} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 270,\ (\ :1/2),\ 0.0762 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945855 - 0.876326i\)
\(L(\frac12)\) \(\approx\) \(0.945855 - 0.876326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.819 - 0.573i)T \)
3 \( 1 + (0.738 + 1.56i)T \)
5 \( 1 + (0.334 + 2.21i)T \)
good7 \( 1 + (1.37 + 2.94i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-0.631 - 0.752i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (1.82 + 2.60i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.889 + 3.31i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.66 + 2.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.99 - 3.72i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-1.14 - 6.46i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-4.97 + 1.81i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (0.374 - 0.100i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.80 + 1.02i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.538 + 6.15i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (7.38 - 3.44i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-8.75 - 8.75i)T + 53iT^{2} \)
59 \( 1 + (-1.28 - 1.07i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.743 - 0.270i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-7.92 + 5.55i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-9.04 - 5.21i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.64 + 1.24i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (12.8 - 2.26i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (4.49 - 6.42i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (1.77 + 3.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.77 + 0.418i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.96855685842188339948004800152, −11.09492997762531149256754558239, −9.755603982792340641501663971883, −8.560235588493999904005006288922, −7.27659480706643591372956528134, −7.04931569705978982104657795118, −5.46278251942146555449873695743, −4.76016974552075406590420553913, −3.12835804429629404216177299705, −0.912810589311304208656116420543, 2.64684612764436729070284254114, 3.60533830358139488119035998965, 4.91138145871148228143536843738, 6.03586063503273697898901084154, 6.75190024710244887994499621521, 8.563331377610103048806980629904, 9.676075301159495078892961091420, 10.27608912501746091388394509039, 11.49656622696355239604361087595, 11.75222407289954619297353772356

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