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Properties

Label 2-273-91.76-c1-0-7
Degree $2$
Conductor $273$
Sign $0.904 + 0.426i$
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  + (−0.706 + 2.63i)2-s + (−0.866 + 0.5i)3-s + (−4.72 − 2.72i)4-s + (−2.18 + 2.18i)5-s + (−0.706 − 2.63i)6-s + (0.666 − 2.56i)7-s + (6.66 − 6.66i)8-s + (0.499 − 0.866i)9-s + (−4.21 − 7.30i)10-s + (−0.456 − 0.122i)11-s + 5.45·12-s + (−2.45 + 2.64i)13-s + (6.28 + 3.56i)14-s + (0.799 − 2.98i)15-s + (7.42 + 12.8i)16-s + (−1.14 + 1.97i)17-s + ⋯
L(s)  = 1  + (−0.499 + 1.86i)2-s + (−0.499 + 0.288i)3-s + (−2.36 − 1.36i)4-s + (−0.976 + 0.976i)5-s + (−0.288 − 1.07i)6-s + (0.251 − 0.967i)7-s + (2.35 − 2.35i)8-s + (0.166 − 0.288i)9-s + (−1.33 − 2.30i)10-s + (−0.137 − 0.0368i)11-s + 1.57·12-s + (−0.680 + 0.732i)13-s + (1.67 + 0.953i)14-s + (0.206 − 0.770i)15-s + (1.85 + 3.21i)16-s + (−0.276 + 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0635211 - 0.0142269i\)
\(L(\frac12)\) \(\approx\) \(0.0635211 - 0.0142269i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.666 + 2.56i)T \)
13 \( 1 + (2.45 - 2.64i)T \)
good2 \( 1 + (0.706 - 2.63i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (2.18 - 2.18i)T - 5iT^{2} \)
11 \( 1 + (0.456 + 0.122i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.14 - 1.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.51 + 5.66i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.481 + 0.278i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.64 + 6.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.74 + 2.74i)T - 31iT^{2} \)
37 \( 1 + (6.41 + 1.71i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.49 + 0.400i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.08 - 2.93i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.55 + 6.55i)T + 47iT^{2} \)
53 \( 1 - 4.17T + 53T^{2} \)
59 \( 1 + (14.2 - 3.82i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.553 - 0.319i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.17 - 8.10i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.13 - 0.572i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.43 + 2.43i)T + 73iT^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (1.80 - 1.80i)T - 83iT^{2} \)
89 \( 1 + (0.363 - 1.35i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.25 + 12.1i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.53434186250766937787712942757, −10.66330504357851664037556646260, −9.810832608208474547131921429039, −8.656195273245175006689486245917, −7.50422294611002428475699379811, −7.09960876245241299949346967800, −6.22089105882563773373898724856, −4.75874189875959779501495188334, −4.03006056231466590996495194512, −0.06511455687431420090505002030, 1.60988940624299815057842857530, 3.14117154382593299882354187839, 4.53482669662289890330201269598, 5.33090289270066212703161454327, 7.69576857848726955917746566693, 8.420141308950108641312539699188, 9.210276652207813751591863683724, 10.32125104429669519213090962833, 11.20878939574244411973046948101, 12.14666290442741628325164336327

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