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Properties

Label 2-273-91.9-c1-0-18
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: $\chi_{273} (100, \cdot )$
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

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

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