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Properties

Label 2-273-91.83-c1-0-1
Degree $2$
Conductor $273$
Sign $-0.590 - 0.806i$
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  + i·3-s + 2i·4-s + (−0.581 − 0.581i)5-s + (−2.58 + 0.581i)7-s − 9-s + (4.16 + 4.16i)11-s − 2·12-s + (−3.58 + 0.418i)13-s + (0.581 − 0.581i)15-s − 4·16-s − 1.16·17-s + (−0.418 − 0.418i)19-s + (1.16 − 1.16i)20-s + (−0.581 − 2.58i)21-s + 4.16i·23-s + ⋯
L(s)  = 1  + 0.577i·3-s + i·4-s + (−0.259 − 0.259i)5-s + (−0.975 + 0.219i)7-s − 0.333·9-s + (1.25 + 1.25i)11-s − 0.577·12-s + (−0.993 + 0.116i)13-s + (0.150 − 0.150i)15-s − 16-s − 0.281·17-s + (−0.0960 − 0.0960i)19-s + (0.259 − 0.259i)20-s + (−0.126 − 0.563i)21-s + 0.867i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.590 - 0.806i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.590 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.430785 + 0.849210i\)
\(L(\frac12)\) \(\approx\) \(0.430785 + 0.849210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (2.58 - 0.581i)T \)
13 \( 1 + (3.58 - 0.418i)T \)
good2 \( 1 - 2iT^{2} \)
5 \( 1 + (0.581 + 0.581i)T + 5iT^{2} \)
11 \( 1 + (-4.16 - 4.16i)T + 11iT^{2} \)
17 \( 1 + 1.16T + 17T^{2} \)
19 \( 1 + (0.418 + 0.418i)T + 19iT^{2} \)
23 \( 1 - 4.16iT - 23T^{2} \)
29 \( 1 - 1.83T + 29T^{2} \)
31 \( 1 + (-5.58 - 5.58i)T + 31iT^{2} \)
37 \( 1 + (-4.32 - 4.32i)T + 37iT^{2} \)
41 \( 1 + (-7.16 - 7.16i)T + 41iT^{2} \)
43 \( 1 + 5.32iT - 43T^{2} \)
47 \( 1 + (-6.58 + 6.58i)T - 47iT^{2} \)
53 \( 1 + 4.16T + 53T^{2} \)
59 \( 1 + (-8.32 + 8.32i)T - 59iT^{2} \)
61 \( 1 - 3.16iT - 61T^{2} \)
67 \( 1 + (6.32 - 6.32i)T - 67iT^{2} \)
71 \( 1 + (-6 + 6i)T - 71iT^{2} \)
73 \( 1 + (5.58 - 5.58i)T - 73iT^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + (-6.58 - 6.58i)T + 83iT^{2} \)
89 \( 1 + (-5.41 + 5.41i)T - 89iT^{2} \)
97 \( 1 + (-0.418 - 0.418i)T + 97iT^{2} \)
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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

−12.14850648273499201954843299500, −11.65830715119000552874286092541, −10.06045340897967449806170020638, −9.424709153811144395430017019299, −8.550246508769274160247298944389, −7.29007512988292688393565735336, −6.48839325982622454478528687318, −4.72041552179378672385096697979, −3.93241317483145620196418602690, −2.63153434223661009255706516467, 0.73515486922434665240353910401, 2.68649122862001342873151825280, 4.19875096733809641258055966341, 5.87011274459756255428959965400, 6.44613244189136516747984825918, 7.43223274598927216856801252440, 8.891706544306241096282994845383, 9.617935996247333949233093671854, 10.71446943867231607471048735763, 11.49934105105994224219186881146

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