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Properties

Label 2-273-91.76-c1-0-5
Degree $2$
Conductor $273$
Sign $0.460 - 0.887i$
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  + (−0.281 + 1.04i)2-s + (0.866 − 0.5i)3-s + (0.710 + 0.410i)4-s + (1.02 − 1.02i)5-s + (0.281 + 1.04i)6-s + (−1.22 + 2.34i)7-s + (−2.16 + 2.16i)8-s + (0.499 − 0.866i)9-s + (0.788 + 1.36i)10-s + (0.972 + 0.260i)11-s + 0.820·12-s + (3.05 + 1.91i)13-s + (−2.11 − 1.94i)14-s + (0.376 − 1.40i)15-s + (−0.842 − 1.45i)16-s + (2.37 − 4.10i)17-s + ⋯
L(s)  = 1  + (−0.198 + 0.741i)2-s + (0.499 − 0.288i)3-s + (0.355 + 0.205i)4-s + (0.459 − 0.459i)5-s + (0.114 + 0.428i)6-s + (−0.462 + 0.886i)7-s + (−0.765 + 0.765i)8-s + (0.166 − 0.288i)9-s + (0.249 + 0.432i)10-s + (0.293 + 0.0785i)11-s + 0.236·12-s + (0.847 + 0.530i)13-s + (−0.565 − 0.519i)14-s + (0.0970 − 0.362i)15-s + (−0.210 − 0.364i)16-s + (0.574 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.460 - 0.887i$
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: \((2,\ 273,\ (\ :1/2),\ 0.460 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32983 + 0.808374i\)
\(L(\frac12)\) \(\approx\) \(1.32983 + 0.808374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 2.34i)T \)
13 \( 1 + (-3.05 - 1.91i)T \)
good2 \( 1 + (0.281 - 1.04i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.02 + 1.02i)T - 5iT^{2} \)
11 \( 1 + (-0.972 - 0.260i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.37 + 4.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.391 + 1.46i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.337 - 0.194i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.30 + 7.44i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.92 - 2.92i)T - 31iT^{2} \)
37 \( 1 + (-9.77 - 2.61i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.64 + 2.31i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (8.28 + 4.78i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.78 + 4.78i)T + 47iT^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 + (0.889 - 0.238i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (8.36 + 4.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.71 + 6.39i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.56 - 0.687i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.46 - 2.46i)T + 73iT^{2} \)
79 \( 1 - 7.06T + 79T^{2} \)
83 \( 1 + (2.43 - 2.43i)T - 83iT^{2} \)
89 \( 1 + (2.79 - 10.4i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.55 + 13.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.01134006015694929421275177249, −11.43988178504605063848156430113, −9.702135908420925755193378494907, −9.016321022258502641151413055100, −8.261939382409357767358016979382, −7.11845410964278972429325438991, −6.24750314848819365728306878609, −5.30764966715980240131170183998, −3.38298449351644260987298909639, −2.05268138488607926584246972654, 1.50259849362265783496924169749, 3.07224513918794230123460256850, 3.88392148030370736967138141278, 5.87125176182888351786144127259, 6.71020055232466383398156079851, 7.954635283011745653417503807683, 9.208294124592292407652433833934, 10.18757805700599259514706056673, 10.51602658344477116101444053127, 11.44804371753231505528104813252

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