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Properties

Label 2-273-13.10-c1-0-5
Degree $2$
Conductor $273$
Sign $0.756 - 0.654i$
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  + (0.954 − 0.551i)2-s + (−0.5 − 0.866i)3-s + (−0.392 + 0.680i)4-s + 3.28i·5-s + (−0.954 − 0.551i)6-s + (0.866 + 0.5i)7-s + 3.06i·8-s + (−0.499 + 0.866i)9-s + (1.80 + 3.13i)10-s + (−1.21 + 0.704i)11-s + 0.785·12-s + (3.42 + 1.13i)13-s + 1.10·14-s + (2.84 − 1.64i)15-s + (0.906 + 1.56i)16-s + (2.88 − 5.00i)17-s + ⋯
L(s)  = 1  + (0.674 − 0.389i)2-s + (−0.288 − 0.499i)3-s + (−0.196 + 0.340i)4-s + 1.46i·5-s + (−0.389 − 0.224i)6-s + (0.327 + 0.188i)7-s + 1.08i·8-s + (−0.166 + 0.288i)9-s + (0.572 + 0.991i)10-s + (−0.367 + 0.212i)11-s + 0.226·12-s + (0.948 + 0.315i)13-s + 0.294·14-s + (0.734 − 0.424i)15-s + (0.226 + 0.392i)16-s + (0.700 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.756 - 0.654i$
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: \((2,\ 273,\ (\ :1/2),\ 0.756 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41567 + 0.527256i\)
\(L(\frac12)\) \(\approx\) \(1.41567 + 0.527256i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.42 - 1.13i)T \)
good2 \( 1 + (-0.954 + 0.551i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 3.28iT - 5T^{2} \)
11 \( 1 + (1.21 - 0.704i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.88 + 5.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.36 + 3.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.70 - 2.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.28 - 7.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.64iT - 31T^{2} \)
37 \( 1 + (-4.84 + 2.79i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.999 - 0.576i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.409 + 0.709i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.35iT - 47T^{2} \)
53 \( 1 - 5.54T + 53T^{2} \)
59 \( 1 + (4.23 + 2.44i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.48 + 2.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.232 + 0.134i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.03 + 4.64i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 - 6.00T + 79T^{2} \)
83 \( 1 - 6.02iT - 83T^{2} \)
89 \( 1 + (-14.3 + 8.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (16.2 + 9.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.91844544915108053668251657337, −11.21623706183911624634967045224, −10.64659940329301678830410347315, −9.123757258415580759019828870134, −7.907132210725283327896248745394, −7.05401827102297794046421367349, −5.97609317563803953080713314559, −4.72549704609056767682240084482, −3.31731039768377144785294118965, −2.35573200798291593166441127509, 1.10124861593729235968586562643, 3.88465582976698602631065516611, 4.62911313619981528091334062384, 5.58946967142636744557479443238, 6.30809161329782675842957180239, 8.202532494472530583911443940499, 8.735181035780929446055134896021, 10.06539652427748852356150846150, 10.66515884904797180961038405089, 12.11155849789862630852461683741

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