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Properties

Label 2-273-91.83-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.390 + 0.920i$
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  + (−1.43 + 1.43i)2-s + i·3-s − 2.13i·4-s + (−0.471 − 0.471i)5-s + (−1.43 − 1.43i)6-s + (−0.645 + 2.56i)7-s + (0.196 + 0.196i)8-s − 9-s + 1.35·10-s + (−3.28 − 3.28i)11-s + 2.13·12-s + (−1.56 + 3.24i)13-s + (−2.76 − 4.61i)14-s + (0.471 − 0.471i)15-s + 3.70·16-s − 5.13·17-s + ⋯
L(s)  = 1  + (−1.01 + 1.01i)2-s + 0.577i·3-s − 1.06i·4-s + (−0.210 − 0.210i)5-s + (−0.587 − 0.587i)6-s + (−0.243 + 0.969i)7-s + (0.0695 + 0.0695i)8-s − 0.333·9-s + 0.428·10-s + (−0.990 − 0.990i)11-s + 0.616·12-s + (−0.433 + 0.901i)13-s + (−0.738 − 1.23i)14-s + (0.121 − 0.121i)15-s + 0.926·16-s − 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.390 + 0.920i$
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: \((2,\ 273,\ (\ :1/2),\ -0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102720 - 0.155080i\)
\(L(\frac12)\) \(\approx\) \(0.102720 - 0.155080i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.645 - 2.56i)T \)
13 \( 1 + (1.56 - 3.24i)T \)
good2 \( 1 + (1.43 - 1.43i)T - 2iT^{2} \)
5 \( 1 + (0.471 + 0.471i)T + 5iT^{2} \)
11 \( 1 + (3.28 + 3.28i)T + 11iT^{2} \)
17 \( 1 + 5.13T + 17T^{2} \)
19 \( 1 + (1.77 + 1.77i)T + 19iT^{2} \)
23 \( 1 - 2.90iT - 23T^{2} \)
29 \( 1 - 7.36T + 29T^{2} \)
31 \( 1 + (0.942 + 0.942i)T + 31iT^{2} \)
37 \( 1 + (3.71 + 3.71i)T + 37iT^{2} \)
41 \( 1 + (-0.744 - 0.744i)T + 41iT^{2} \)
43 \( 1 - 8.69iT - 43T^{2} \)
47 \( 1 + (3.25 - 3.25i)T - 47iT^{2} \)
53 \( 1 + 5.78T + 53T^{2} \)
59 \( 1 + (5.95 - 5.95i)T - 59iT^{2} \)
61 \( 1 - 13.0iT - 61T^{2} \)
67 \( 1 + (7.50 - 7.50i)T - 67iT^{2} \)
71 \( 1 + (8.19 - 8.19i)T - 71iT^{2} \)
73 \( 1 + (-10.2 + 10.2i)T - 73iT^{2} \)
79 \( 1 - 2.47T + 79T^{2} \)
83 \( 1 + (-10.0 - 10.0i)T + 83iT^{2} \)
89 \( 1 + (-0.170 + 0.170i)T - 89iT^{2} \)
97 \( 1 + (11.3 + 11.3i)T + 97iT^{2} \)
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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

−12.44491593821656328033205474707, −11.39622965010876869987023245374, −10.37554119896048548592328885129, −9.294391173156325397864587028453, −8.745679453063438580158048855376, −7.998886601018372974905691610460, −6.66924490562711758547538694548, −5.82724475774831153651582426369, −4.59522528846090702186404828233, −2.77586996369200262290447196106, 0.18460835000804526776370452342, 1.99359190165056100775719988289, 3.20348310789185025814486715918, 4.89315537626414401733612790023, 6.63448670290718633246827612852, 7.61509526435333757646210871167, 8.360681070936537667862361049627, 9.605125530220178884903965757625, 10.52196016424988893788993123097, 10.83971777470765744383655825210

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