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Properties

Label 2-273-91.59-c1-0-10
Degree $2$
Conductor $273$
Sign $0.721 + 0.692i$
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  + (−0.240 − 0.240i)2-s + (0.866 − 0.5i)3-s − 1.88i·4-s + (2.06 + 0.552i)5-s + (−0.328 − 0.0881i)6-s + (1.35 + 2.27i)7-s + (−0.935 + 0.935i)8-s + (0.499 − 0.866i)9-s + (−0.363 − 0.629i)10-s + (0.208 + 0.0558i)11-s + (−0.942 − 1.63i)12-s + (3.58 − 0.408i)13-s + (0.219 − 0.873i)14-s + (2.06 − 0.552i)15-s − 3.31·16-s − 5.29·17-s + ⋯
L(s)  = 1  + (−0.170 − 0.170i)2-s + (0.499 − 0.288i)3-s − 0.942i·4-s + (0.922 + 0.247i)5-s + (−0.134 − 0.0359i)6-s + (0.513 + 0.858i)7-s + (−0.330 + 0.330i)8-s + (0.166 − 0.288i)9-s + (−0.115 − 0.199i)10-s + (0.0628 + 0.0168i)11-s + (−0.271 − 0.471i)12-s + (0.993 − 0.113i)13-s + (0.0586 − 0.233i)14-s + (0.532 − 0.142i)15-s − 0.829·16-s − 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.721 + 0.692i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47414 - 0.593365i\)
\(L(\frac12)\) \(\approx\) \(1.47414 - 0.593365i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 - 2.27i)T \)
13 \( 1 + (-3.58 + 0.408i)T \)
good2 \( 1 + (0.240 + 0.240i)T + 2iT^{2} \)
5 \( 1 + (-2.06 - 0.552i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.208 - 0.0558i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 + (1.66 + 6.21i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 6.44iT - 23T^{2} \)
29 \( 1 + (-1.21 + 2.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.681 + 2.54i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.73 + 2.73i)T - 37iT^{2} \)
41 \( 1 + (-2.28 - 8.53i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.95 - 2.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.31 - 12.3i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.97 + 3.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.64 + 8.64i)T + 59iT^{2} \)
61 \( 1 + (2.44 + 1.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.436 + 1.62i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.653 + 2.44i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.0329 + 0.00884i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.28 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.72 - 7.72i)T - 83iT^{2} \)
89 \( 1 + (5.91 + 5.91i)T + 89iT^{2} \)
97 \( 1 + (-7.90 - 2.11i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.37169016002893609605827407857, −11.03824810395044494376875410923, −9.586833353766454561603476267505, −9.198881203710442253026456153229, −8.152919623700231969531805607926, −6.54829782880853525719902993387, −5.92823126124473062597019561705, −4.70308516016205461122547209199, −2.63569869123099725134945060704, −1.65743238946840188733890506584, 1.97033156123213806063625163106, 3.63143399785175480919402041548, 4.55708864628115781596515461513, 6.18623273844573251033401746491, 7.22322272572832004629430552076, 8.455478089210199033181196307600, 8.806693779442631574470698392980, 10.13089337789431820754381658667, 10.87268105657781383477300976389, 12.11037008230086048660750987643

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