Properties

Label 2-273-91.83-c1-0-17
Degree $2$
Conductor $273$
Sign $-0.630 + 0.776i$
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  + (1.22 − 1.22i)2-s i·3-s − 0.999i·4-s + (−2 − 2i)5-s + (−1.22 − 1.22i)6-s + (1 − 2.44i)7-s + (1.22 + 1.22i)8-s − 9-s − 4.89·10-s + (−0.449 − 0.449i)11-s − 0.999·12-s + (−2 + 3i)13-s + (−1.77 − 4.22i)14-s + (−2 + 2i)15-s + 5·16-s + 2·17-s + ⋯
L(s)  = 1  + (0.866 − 0.866i)2-s − 0.577i·3-s − 0.499i·4-s + (−0.894 − 0.894i)5-s + (−0.499 − 0.499i)6-s + (0.377 − 0.925i)7-s + (0.433 + 0.433i)8-s − 0.333·9-s − 1.54·10-s + (−0.135 − 0.135i)11-s − 0.288·12-s + (−0.554 + 0.832i)13-s + (−0.474 − 1.12i)14-s + (−0.516 + 0.516i)15-s + 1.25·16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.630 + 0.776i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.630 + 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752176 - 1.57883i\)
\(L(\frac12)\) \(\approx\) \(0.752176 - 1.57883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (-1 + 2.44i)T \)
13 \( 1 + (2 - 3i)T \)
good2 \( 1 + (-1.22 + 1.22i)T - 2iT^{2} \)
5 \( 1 + (2 + 2i)T + 5iT^{2} \)
11 \( 1 + (0.449 + 0.449i)T + 11iT^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (-0.550 - 0.550i)T + 19iT^{2} \)
23 \( 1 + 8.89iT - 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 + (-5.44 - 5.44i)T + 31iT^{2} \)
37 \( 1 + (-7.89 - 7.89i)T + 37iT^{2} \)
41 \( 1 + (-4 - 4i)T + 41iT^{2} \)
43 \( 1 - 6.89iT - 43T^{2} \)
47 \( 1 + (1.55 - 1.55i)T - 47iT^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + (-4.44 + 4.44i)T - 59iT^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 + (6.34 - 6.34i)T - 67iT^{2} \)
71 \( 1 + (-2.44 + 2.44i)T - 71iT^{2} \)
73 \( 1 + (7.89 - 7.89i)T - 73iT^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 + (0.449 + 0.449i)T + 83iT^{2} \)
89 \( 1 + (2 - 2i)T - 89iT^{2} \)
97 \( 1 + (7.89 + 7.89i)T + 97iT^{2} \)
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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

−11.78335153007984385847714154544, −11.09167790581676379323532207270, −9.966969438494112775108364282996, −8.329737683479690016816681381942, −7.87963834668404501310736882369, −6.57197364372148838592487478230, −4.72416777117522853757758542864, −4.41707444395358256015728101616, −2.91523800453843093926981502307, −1.18640937313623032494966389139, 2.93464523081225141131066306959, 4.07671566487956019322798460834, 5.25441587395853182134093092199, 5.98241657034860454174584029530, 7.40554456387400213538618322281, 7.893673005435995003922091007169, 9.434808133263976059364051279990, 10.43403960873768318342670385157, 11.42871477903848217062597867730, 12.22126180067381340257147523633

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