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Properties

Label 2-273-13.10-c1-0-12
Degree $2$
Conductor $273$
Sign $0.672 + 0.740i$
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  + (1.85 − 1.07i)2-s + (0.5 + 0.866i)3-s + (1.30 − 2.25i)4-s − 1.91i·5-s + (1.85 + 1.07i)6-s + (0.866 + 0.5i)7-s − 1.29i·8-s + (−0.499 + 0.866i)9-s + (−2.05 − 3.56i)10-s + (−1.23 + 0.710i)11-s + 2.60·12-s + (1.05 − 3.44i)13-s + 2.14·14-s + (1.66 − 0.959i)15-s + (1.21 + 2.10i)16-s + (−3.23 + 5.60i)17-s + ⋯
L(s)  = 1  + (1.31 − 0.758i)2-s + (0.288 + 0.499i)3-s + (0.651 − 1.12i)4-s − 0.858i·5-s + (0.758 + 0.438i)6-s + (0.327 + 0.188i)7-s − 0.458i·8-s + (−0.166 + 0.288i)9-s + (−0.651 − 1.12i)10-s + (−0.371 + 0.214i)11-s + 0.751·12-s + (0.292 − 0.956i)13-s + 0.573·14-s + (0.429 − 0.247i)15-s + (0.303 + 0.525i)16-s + (−0.784 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.672 + 0.740i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.672 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38072 - 1.05363i\)
\(L(\frac12)\) \(\approx\) \(2.38072 - 1.05363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-1.05 + 3.44i)T \)
good2 \( 1 + (-1.85 + 1.07i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 1.91iT - 5T^{2} \)
11 \( 1 + (1.23 - 0.710i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.23 - 5.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.97 + 2.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.07 + 1.85i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.81 - 6.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.50iT - 31T^{2} \)
37 \( 1 + (6.77 - 3.91i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.52 - 4.34i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.53 + 6.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.362iT - 47T^{2} \)
53 \( 1 - 0.524T + 53T^{2} \)
59 \( 1 + (7.80 + 4.50i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.95 + 6.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.351 - 0.202i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.210 - 0.121i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.330iT - 73T^{2} \)
79 \( 1 - 8.64T + 79T^{2} \)
83 \( 1 + 11.4iT - 83T^{2} \)
89 \( 1 + (-10.9 + 6.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.8 - 8.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.05884967337277733944006559777, −10.78628031722035003954086020966, −10.45146386794567552392536973574, −8.764731209374178590422710703234, −8.281706707491622268815301252131, −6.33519135689583522897638165617, −5.11411009125614862013125297358, −4.55085752609710097973871392666, −3.36463871985084796816680617486, −1.98751999059164158085736050719, 2.44775914028182781655924382732, 3.75586862618827002313314807185, 4.86430925450513092523331685465, 6.21324116184455428210162217995, 6.84151632633367455181352593805, 7.66970091758985957581126798539, 8.894328474685893350600728649023, 10.33045231296767621389099998938, 11.43053150144218801145430929379, 12.22639795817328552870667100174

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