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Properties

Label 2-273-91.45-c1-0-9
Degree $2$
Conductor $273$
Sign $0.884 + 0.467i$
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  + (1.09 − 1.09i)2-s + (−0.866 + 0.5i)3-s − 0.399i·4-s + (0.128 − 0.481i)5-s + (−0.400 + 1.49i)6-s + (2.51 + 0.807i)7-s + (1.75 + 1.75i)8-s + (0.499 − 0.866i)9-s + (−0.385 − 0.668i)10-s + (0.483 − 1.80i)11-s + (0.199 + 0.345i)12-s + (3.48 − 0.924i)13-s + (3.64 − 1.87i)14-s + (0.128 + 0.481i)15-s + 4.63·16-s − 3.91·17-s + ⋯
L(s)  = 1  + (0.774 − 0.774i)2-s + (−0.499 + 0.288i)3-s − 0.199i·4-s + (0.0576 − 0.215i)5-s + (−0.163 + 0.610i)6-s + (0.952 + 0.305i)7-s + (0.619 + 0.619i)8-s + (0.166 − 0.288i)9-s + (−0.122 − 0.211i)10-s + (0.145 − 0.544i)11-s + (0.0576 + 0.0997i)12-s + (0.966 − 0.256i)13-s + (0.973 − 0.501i)14-s + (0.0333 + 0.124i)15-s + 1.15·16-s − 0.950·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.884 + 0.467i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.884 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76245 - 0.437026i\)
\(L(\frac12)\) \(\approx\) \(1.76245 - 0.437026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 - 0.807i)T \)
13 \( 1 + (-3.48 + 0.924i)T \)
good2 \( 1 + (-1.09 + 1.09i)T - 2iT^{2} \)
5 \( 1 + (-0.128 + 0.481i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.483 + 1.80i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 3.91T + 17T^{2} \)
19 \( 1 + (3.69 - 0.991i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.91iT - 23T^{2} \)
29 \( 1 + (2.28 - 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.21 - 1.12i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.106 - 0.106i)T + 37iT^{2} \)
41 \( 1 + (4.92 - 1.31i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.06 - 2.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.92 + 2.39i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.56 + 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.69 + 2.69i)T - 59iT^{2} \)
61 \( 1 + (10.1 + 5.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.55 + 2.56i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.82 + 0.487i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.75 - 6.55i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.22 - 3.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.85 - 5.85i)T + 83iT^{2} \)
89 \( 1 + (-8.74 + 8.74i)T - 89iT^{2} \)
97 \( 1 + (-2.09 + 7.81i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.71167377431584618952903756949, −11.05237321536226092954359740116, −10.55730329929926373629066531636, −8.835410333458766290966091586869, −8.233491419871916608729310687911, −6.59784830651589149909971338175, −5.33790209010175753932920347490, −4.55066060219102934357238902908, −3.40938751995027564414235829910, −1.76986492654352566932191521738, 1.67537052213803472741668885442, 4.05895928701140431616637444182, 4.88366076919930523689278482532, 6.00939616298721180588025355215, 6.82756536487436840355874400286, 7.68383609505939131619760876324, 8.937487003048187348864611107104, 10.40674750396327008867894367987, 11.05528345848827005441861140138, 12.03947794410919529152097617291

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