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Properties

Label 2-273-91.73-c1-0-12
Degree $2$
Conductor $273$
Sign $0.676 + 0.736i$
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  + (0.345 − 1.28i)2-s + (0.866 + 0.5i)3-s + (0.191 + 0.110i)4-s + (1.14 + 0.306i)5-s + (0.943 − 0.943i)6-s + (−2.38 − 1.14i)7-s + (2.09 − 2.09i)8-s + (0.499 + 0.866i)9-s + (0.791 − 1.37i)10-s + (0.778 + 2.90i)11-s + (0.110 + 0.191i)12-s + (0.759 − 3.52i)13-s + (−2.30 + 2.67i)14-s + (0.838 + 0.838i)15-s + (−1.75 − 3.03i)16-s + (−0.562 + 0.974i)17-s + ⋯
L(s)  = 1  + (0.244 − 0.911i)2-s + (0.499 + 0.288i)3-s + (0.0955 + 0.0551i)4-s + (0.512 + 0.137i)5-s + (0.385 − 0.385i)6-s + (−0.901 − 0.433i)7-s + (0.740 − 0.740i)8-s + (0.166 + 0.288i)9-s + (0.250 − 0.433i)10-s + (0.234 + 0.876i)11-s + (0.0318 + 0.0551i)12-s + (0.210 − 0.977i)13-s + (−0.615 + 0.715i)14-s + (0.216 + 0.216i)15-s + (−0.438 − 0.759i)16-s + (−0.136 + 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72495 - 0.758011i\)
\(L(\frac12)\) \(\approx\) \(1.72495 - 0.758011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (2.38 + 1.14i)T \)
13 \( 1 + (-0.759 + 3.52i)T \)
good2 \( 1 + (-0.345 + 1.28i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.14 - 0.306i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.778 - 2.90i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.562 - 0.974i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.38 - 1.70i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.76 - 2.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.66T + 29T^{2} \)
31 \( 1 + (-0.678 - 2.53i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (9.55 + 2.56i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.47 - 3.47i)T - 41iT^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + (1.07 - 4.00i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.28 - 7.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.06 + 2.43i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (8.03 - 4.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.31 - 0.888i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.61 - 7.61i)T + 71iT^{2} \)
73 \( 1 + (-10.6 + 2.84i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.95 - 5.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0499 - 0.0499i)T - 83iT^{2} \)
89 \( 1 + (-3.65 + 13.6i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.73 + 2.73i)T - 97iT^{2} \)
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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

−11.91591109893879185550774744338, −10.62118202298399999188304197583, −10.02085647250317625212578021893, −9.426377087261808481494956532950, −7.80567073914587357734052393362, −6.96959431103196475950886768665, −5.58120790327924159121769872983, −3.93844536458336687154746612429, −3.19530296408252260224310462602, −1.82239844738568061026125853385, 1.98848847741060385705710578601, 3.52798650154681429788830885240, 5.26811382167089251905094258848, 6.19602375876556557245718066692, 6.88504070734316269700021556574, 8.018068257709965285227079550512, 9.091183756559177983605843717002, 9.803327840645523073865003842514, 11.22705661563284126161857549517, 12.07280679166067710434831740135

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