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Properties

Label 2-273-91.81-c1-0-17
Degree $2$
Conductor $273$
Sign $-0.991 - 0.129i$
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  + (1.21 − 2.10i)2-s − 3-s + (−1.96 − 3.39i)4-s + (−0.613 − 1.06i)5-s + (−1.21 + 2.10i)6-s + (−2.37 − 1.17i)7-s − 4.68·8-s + 9-s − 2.98·10-s + 3.49·11-s + (1.96 + 3.39i)12-s + (−2.87 + 2.17i)13-s + (−5.35 + 3.57i)14-s + (0.613 + 1.06i)15-s + (−1.77 + 3.07i)16-s + (−2.26 − 3.92i)17-s + ⋯
L(s)  = 1  + (0.860 − 1.49i)2-s − 0.577·3-s + (−0.981 − 1.69i)4-s + (−0.274 − 0.475i)5-s + (−0.496 + 0.860i)6-s + (−0.896 − 0.443i)7-s − 1.65·8-s + 0.333·9-s − 0.945·10-s + 1.05·11-s + (0.566 + 0.981i)12-s + (−0.797 + 0.603i)13-s + (−1.43 + 0.954i)14-s + (0.158 + 0.274i)15-s + (−0.444 + 0.769i)16-s + (−0.548 − 0.950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.991 - 0.129i$
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: \((2,\ 273,\ (\ :1/2),\ -0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0827869 + 1.26847i\)
\(L(\frac12)\) \(\approx\) \(0.0827869 + 1.26847i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.37 + 1.17i)T \)
13 \( 1 + (2.87 - 2.17i)T \)
good2 \( 1 + (-1.21 + 2.10i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.613 + 1.06i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.49T + 11T^{2} \)
17 \( 1 + (2.26 + 3.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.35T + 19T^{2} \)
23 \( 1 + (-0.336 + 0.583i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.64 - 4.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.99 + 8.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.54 + 2.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.61 - 6.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.48 + 7.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.58 - 4.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.95 - 8.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.401 + 0.695i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.64T + 61T^{2} \)
67 \( 1 + 2.12T + 67T^{2} \)
71 \( 1 + (2.52 - 4.37i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.04 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.90 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + (-1.55 + 2.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.59 - 6.22i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.69821165415246062650217741518, −10.76502573963215233944256178485, −9.738555302424986412496913525707, −9.172407621040815337671854614026, −7.20490769494980618852657562851, −6.09744070402600043472952388260, −4.68138746757880391794788485674, −4.11257827813033024880748788040, −2.66988838248982405536375533956, −0.836061763690352632054931884728, 3.24898522566914045487113747289, 4.41910681068839580786354986638, 5.59041303349842580841327634603, 6.47929304737393186369037930138, 7.00280297243859914496074637709, 8.182257924446248105515370354636, 9.333251895956782830955081626307, 10.52979277213326627841809160868, 11.88147169361276542979546143512, 12.55061600242567760207458828006

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