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Properties

Label 2-273-91.81-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.367 - 0.930i$
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  + (−0.0340 + 0.0589i)2-s − 3-s + (0.997 + 1.72i)4-s + (1.52 + 2.64i)5-s + (0.0340 − 0.0589i)6-s + (−2.60 + 0.453i)7-s − 0.271·8-s + 9-s − 0.208·10-s − 4.35·11-s + (−0.997 − 1.72i)12-s + (−1.79 − 3.12i)13-s + (0.0619 − 0.169i)14-s + (−1.52 − 2.64i)15-s + (−1.98 + 3.44i)16-s + (1.76 + 3.05i)17-s + ⋯
L(s)  = 1  + (−0.0240 + 0.0416i)2-s − 0.577·3-s + (0.498 + 0.864i)4-s + (0.684 + 1.18i)5-s + (0.0138 − 0.0240i)6-s + (−0.985 + 0.171i)7-s − 0.0961·8-s + 0.333·9-s − 0.0658·10-s − 1.31·11-s + (−0.288 − 0.498i)12-s + (−0.499 − 0.866i)13-s + (0.0165 − 0.0451i)14-s + (−0.394 − 0.684i)15-s + (−0.496 + 0.860i)16-s + (0.427 + 0.741i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.367 - 0.930i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.367 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577969 + 0.849716i\)
\(L(\frac12)\) \(\approx\) \(0.577969 + 0.849716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (2.60 - 0.453i)T \)
13 \( 1 + (1.79 + 3.12i)T \)
good2 \( 1 + (0.0340 - 0.0589i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.52 - 2.64i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 4.35T + 11T^{2} \)
17 \( 1 + (-1.76 - 3.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.90T + 19T^{2} \)
23 \( 1 + (1.66 - 2.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.95 - 8.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.62 + 8.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0545 + 0.0944i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.76 - 3.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.844 - 1.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.28 - 2.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.65 + 4.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.77 + 6.54i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.87T + 61T^{2} \)
67 \( 1 - 0.680T + 67T^{2} \)
71 \( 1 + (2.61 - 4.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.75 + 3.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.85 - 8.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.41T + 83T^{2} \)
89 \( 1 + (-3.85 + 6.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.86 - 6.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.26270062818366870356585751244, −11.18043652921814508234071411656, −10.29333502883538968229422695475, −9.761834369431343883460293034664, −8.005043178537110888167147195933, −7.22787818453764060850884750714, −6.28287135156570822498059643582, −5.40807695798457664556952866843, −3.32056686032733198577577494538, −2.63779171318554139781172850419, 0.838805646837538446624014371723, 2.57148105726743112264263825145, 4.80054639106317520791016108823, 5.45712521719132133721357332857, 6.41900750985475990582515784214, 7.50355340277563898003674340467, 9.102358946063037506613496377147, 9.910958790982005403620213359723, 10.34875078191257731322059543811, 11.81588289296362632253582656164

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