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Properties

Label 2-273-91.41-c1-0-9
Degree $2$
Conductor $273$
Sign $0.626 - 0.779i$
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  + (1.96 + 0.525i)2-s + (−0.866 + 0.5i)3-s + (1.84 + 1.06i)4-s + (1.56 + 1.56i)5-s + (−1.96 + 0.525i)6-s + (2.02 − 1.70i)7-s + (0.187 + 0.187i)8-s + (0.499 − 0.866i)9-s + (2.25 + 3.90i)10-s + (−1.16 + 4.33i)11-s − 2.13·12-s + (−3.51 + 0.786i)13-s + (4.87 − 2.27i)14-s + (−2.14 − 0.573i)15-s + (−1.86 − 3.22i)16-s + (1.31 − 2.27i)17-s + ⋯
L(s)  = 1  + (1.38 + 0.371i)2-s + (−0.499 + 0.288i)3-s + (0.922 + 0.532i)4-s + (0.700 + 0.700i)5-s + (−0.801 + 0.214i)6-s + (0.765 − 0.643i)7-s + (0.0661 + 0.0661i)8-s + (0.166 − 0.288i)9-s + (0.712 + 1.23i)10-s + (−0.350 + 1.30i)11-s − 0.614·12-s + (−0.975 + 0.218i)13-s + (1.30 − 0.609i)14-s + (−0.552 − 0.148i)15-s + (−0.465 − 0.805i)16-s + (0.318 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15513 + 1.03236i\)
\(L(\frac12)\) \(\approx\) \(2.15513 + 1.03236i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 + 1.70i)T \)
13 \( 1 + (3.51 - 0.786i)T \)
good2 \( 1 + (-1.96 - 0.525i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.56 - 1.56i)T + 5iT^{2} \)
11 \( 1 + (1.16 - 4.33i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.31 + 2.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.01 + 1.61i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.58 - 2.64i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.06 + 3.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.44 + 2.44i)T + 31iT^{2} \)
37 \( 1 + (-0.0290 + 0.108i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.03 + 7.58i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.47 + 3.16i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.82 - 5.82i)T - 47iT^{2} \)
53 \( 1 - 9.24T + 53T^{2} \)
59 \( 1 + (-1.27 - 4.77i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.33 + 1.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.1 - 2.72i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.56 - 13.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (11.8 - 11.8i)T - 73iT^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 + (-5.15 - 5.15i)T + 83iT^{2} \)
89 \( 1 + (-7.93 - 2.12i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.93 - 0.518i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.04515493993550936713538320818, −11.46537055638381736766079701103, −10.07010113424732286278689398714, −9.694429025455858873509440371705, −7.44278499735365580968923460751, −7.03376656248193247302778194650, −5.65660546490820686962068901678, −4.97777468346294954628942065029, −3.99119929511605375667997649136, −2.38793767213597525070182325936, 1.77086811832289655966851983600, 3.22197408056256453416228030992, 4.90370722809489386409041147720, 5.42436105186123816574114497193, 6.12056539230552895608089446329, 7.84904878011570252527678173470, 8.853365901165179121433731933684, 10.17449205148119830051890265173, 11.32531531000104866510816824669, 11.95273069058714734122989316154

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