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Properties

Label 2-273-39.32-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.959 - 0.280i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
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  + (−0.751 + 0.201i)2-s + (−1.18 + 1.26i)3-s + (−1.20 + 0.697i)4-s + (2.48 + 2.48i)5-s + (0.633 − 1.18i)6-s + (−0.258 + 0.965i)7-s + (1.86 − 1.86i)8-s + (−0.202 − 2.99i)9-s + (−2.36 − 1.36i)10-s + (1.16 + 4.33i)11-s + (0.546 − 2.35i)12-s + (−2.59 + 2.50i)13-s − 0.777i·14-s + (−6.07 + 0.205i)15-s + (0.367 − 0.636i)16-s + (−2.53 − 4.38i)17-s + ⋯
L(s)  = 1  + (−0.531 + 0.142i)2-s + (−0.682 + 0.730i)3-s + (−0.603 + 0.348i)4-s + (1.10 + 1.10i)5-s + (0.258 − 0.485i)6-s + (−0.0978 + 0.365i)7-s + (0.660 − 0.660i)8-s + (−0.0675 − 0.997i)9-s + (−0.747 − 0.431i)10-s + (0.350 + 1.30i)11-s + (0.157 − 0.679i)12-s + (−0.719 + 0.694i)13-s − 0.207i·14-s + (−1.56 + 0.0530i)15-s + (0.0919 − 0.159i)16-s + (−0.613 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.959 - 0.280i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.959 - 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0899808 + 0.628482i\)
\(L(\frac12)\) \(\approx\) \(0.0899808 + 0.628482i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.18 - 1.26i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (2.59 - 2.50i)T \)
good2 \( 1 + (0.751 - 0.201i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-2.48 - 2.48i)T + 5iT^{2} \)
11 \( 1 + (-1.16 - 4.33i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.53 + 4.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.15 + 1.11i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.39 + 2.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.156 + 0.0905i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.69 - 6.69i)T - 31iT^{2} \)
37 \( 1 + (-7.89 + 2.11i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.61 + 1.50i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.515 + 0.297i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.599 - 0.599i)T - 47iT^{2} \)
53 \( 1 + 5.43iT - 53T^{2} \)
59 \( 1 + (-7.75 - 2.07i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.75 - 3.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.450 + 1.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.53 - 13.1i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.693 - 0.693i)T + 73iT^{2} \)
79 \( 1 - 0.480T + 79T^{2} \)
83 \( 1 + (-8.15 - 8.15i)T + 83iT^{2} \)
89 \( 1 + (-0.837 - 3.12i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.19 + 0.588i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.27867404887366211161045225279, −11.10742704789660440813465585874, −10.20835028238919018314065908502, −9.499581694241883406632437526880, −9.044879183690350030674438483439, −7.12060075397297558608561091452, −6.62929633047417852549688516664, −5.16756156265821174778925880163, −4.19204058534793495066890168716, −2.43395273341722867721688430559, 0.64350840116763394829282176428, 1.88405851543295153593593198498, 4.45480654635077508994420525380, 5.62771970141306925292678477662, 6.10887002883267765846036984089, 7.83435418975245379994180037981, 8.661178690572234719896007768474, 9.535211133741395311278726891095, 10.51902338408899608136269286658, 11.25437553657726729074187237386

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