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Properties

Label 2-273-39.11-c1-0-22
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: $\chi_{273} (50, \cdot )$
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

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

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