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Properties

Label 2-273-91.6-c1-0-7
Degree $2$
Conductor $273$
Sign $0.989 - 0.147i$
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  + (0.0578 + 0.215i)2-s + (−0.866 − 0.5i)3-s + (1.68 − 0.975i)4-s + (1.08 + 1.08i)5-s + (0.0578 − 0.215i)6-s + (−0.725 + 2.54i)7-s + (0.624 + 0.624i)8-s + (0.499 + 0.866i)9-s + (−0.171 + 0.297i)10-s + (4.17 − 1.11i)11-s − 1.95·12-s + (−3.01 − 1.98i)13-s + (−0.591 − 0.00954i)14-s + (−0.398 − 1.48i)15-s + (1.85 − 3.20i)16-s + (3.78 + 6.55i)17-s + ⋯
L(s)  = 1  + (0.0408 + 0.152i)2-s + (−0.499 − 0.288i)3-s + (0.844 − 0.487i)4-s + (0.486 + 0.486i)5-s + (0.0236 − 0.0880i)6-s + (−0.274 + 0.961i)7-s + (0.220 + 0.220i)8-s + (0.166 + 0.288i)9-s + (−0.0543 + 0.0941i)10-s + (1.25 − 0.337i)11-s − 0.562·12-s + (−0.835 − 0.550i)13-s + (−0.157 − 0.00255i)14-s + (−0.102 − 0.383i)15-s + (0.462 − 0.801i)16-s + (0.917 + 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.989 - 0.147i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.989 - 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45529 + 0.107990i\)
\(L(\frac12)\) \(\approx\) \(1.45529 + 0.107990i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.725 - 2.54i)T \)
13 \( 1 + (3.01 + 1.98i)T \)
good2 \( 1 + (-0.0578 - 0.215i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-1.08 - 1.08i)T + 5iT^{2} \)
11 \( 1 + (-4.17 + 1.11i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.78 - 6.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.31 + 4.91i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.85 + 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.66 - 4.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.09 - 1.09i)T + 31iT^{2} \)
37 \( 1 + (2.82 - 0.757i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.37 + 0.369i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.58 - 3.80i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.26 + 5.26i)T - 47iT^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + (8.73 + 2.34i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (10.1 - 5.88i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.28 + 4.78i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.25 + 1.40i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-8.04 + 8.04i)T - 73iT^{2} \)
79 \( 1 - 9.06T + 79T^{2} \)
83 \( 1 + (-1.13 - 1.13i)T + 83iT^{2} \)
89 \( 1 + (0.217 + 0.811i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.35 - 16.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.03232138803184285768350061020, −10.96666511456079263540573968613, −10.21282366287693205635106625059, −9.209885915690839241882561620708, −7.84563966726000858568390561600, −6.57476850976317102337058388117, −6.16487797148387920434759996634, −5.16616986649935521953543938079, −3.07990985947668178752158287212, −1.72566780237312401278477804267, 1.53193350946140344500567295527, 3.42160803515362193926916079966, 4.54260731829600922732151412020, 5.92640172258634929165804203075, 6.99586986227857708749340699694, 7.68439052925882696910449998508, 9.503724035240610766623343901499, 9.822552947422091354494541298458, 11.09834550174688707252587145151, 11.96554385196086753987912475869

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