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Properties

Label 2-273-91.30-c1-0-17
Degree $2$
Conductor $273$
Sign $0.313 + 0.949i$
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  + (1.89 − 1.09i)2-s + 3-s + (1.39 − 2.41i)4-s + (−1.5 − 0.866i)5-s + (1.89 − 1.09i)6-s − 2.64i·7-s − 1.73i·8-s + 9-s − 3.79·10-s + 3.46i·11-s + (1.39 − 2.41i)12-s + (−1 + 3.46i)13-s + (−2.89 − 5.01i)14-s + (−1.5 − 0.866i)15-s + (0.895 + 1.55i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (1.34 − 0.773i)2-s + 0.577·3-s + (0.697 − 1.20i)4-s + (−0.670 − 0.387i)5-s + (0.773 − 0.446i)6-s − 0.999i·7-s − 0.612i·8-s + 0.333·9-s − 1.19·10-s + 1.04i·11-s + (0.402 − 0.697i)12-s + (−0.277 + 0.960i)13-s + (−0.773 − 1.34i)14-s + (−0.387 − 0.223i)15-s + (0.223 + 0.387i)16-s + (−0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.313 + 0.949i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.313 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13900 - 1.54656i\)
\(L(\frac12)\) \(\approx\) \(2.13900 - 1.54656i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64iT \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + (-1.89 + 1.09i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 + (-4.29 - 7.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.29 - 3.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.08 - 3.51i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.08 + 1.77i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.29 + 3.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.708 - 0.409i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.08 - 5.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.70 + 2.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 5.16T + 61T^{2} \)
67 \( 1 + 14.0iT - 67T^{2} \)
71 \( 1 + (3.87 - 2.23i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.5 + 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.708 + 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.08 + 5.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.90878032330958708740923166745, −11.11033386025035541931956534291, −10.06625388637535526905707591159, −8.978367884376287918664782475203, −7.61673933825580041068962048239, −6.77698578274062337517441281903, −4.92072686561926615260449599163, −4.32204017092603312116165338563, −3.36705152562684046556611750721, −1.83294260438404159532570875421, 2.89089078097043507743768710767, 3.62068152088635527120663925246, 5.04634307684214291959330798429, 5.93012579508360195881924611817, 7.02155404160861542453534409790, 8.031651163346810967388474299607, 8.828552673178671109318066264746, 10.33667196805598329626719033604, 11.49782132712552666125144592769, 12.49499796439823967643185620001

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