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Properties

Label 2-273-91.6-c1-0-4
Degree $2$
Conductor $273$
Sign $-0.673 - 0.738i$
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  + (0.5 + 1.86i)2-s + (0.866 + 0.5i)3-s + (−1.5 + 0.866i)4-s + (−0.366 − 0.366i)5-s + (−0.5 + 1.86i)6-s + (−0.866 + 2.5i)7-s + (0.366 + 0.366i)8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (−1 + 0.267i)11-s − 1.73·12-s + (−0.232 − 3.59i)13-s + (−5.09 − 0.366i)14-s + (−0.133 − 0.5i)15-s + (−2.23 + 3.86i)16-s + (2.59 + 4.5i)17-s + ⋯
L(s)  = 1  + (0.353 + 1.31i)2-s + (0.499 + 0.288i)3-s + (−0.750 + 0.433i)4-s + (−0.163 − 0.163i)5-s + (−0.204 + 0.761i)6-s + (−0.327 + 0.944i)7-s + (0.129 + 0.129i)8-s + (0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.301 + 0.0807i)11-s − 0.499·12-s + (−0.0643 − 0.997i)13-s + (−1.36 − 0.0978i)14-s + (−0.0345 − 0.129i)15-s + (−0.558 + 0.966i)16-s + (0.630 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.673 - 0.738i$
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: \((2,\ 273,\ (\ :1/2),\ -0.673 - 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687423 + 1.55753i\)
\(L(\frac12)\) \(\approx\) \(0.687423 + 1.55753i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 2.5i)T \)
13 \( 1 + (0.232 + 3.59i)T \)
good2 \( 1 + (-0.5 - 1.86i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.366 + 0.366i)T + 5iT^{2} \)
11 \( 1 + (1 - 0.267i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.59 - 4.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.732 + 2.73i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.63 - 2.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.73 + 6.73i)T + 31iT^{2} \)
37 \( 1 + (-2.5 + 0.669i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.59 + 2.03i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.92 + 4i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.46 - 9.46i)T - 47iT^{2} \)
53 \( 1 - 4.26T + 53T^{2} \)
59 \( 1 + (-5.09 - 1.36i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.401 + 0.232i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.56 - 5.83i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.46 + 1.46i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (7.29 - 7.29i)T - 73iT^{2} \)
79 \( 1 + 1.66T + 79T^{2} \)
83 \( 1 + (6.92 + 6.92i)T + 83iT^{2} \)
89 \( 1 + (1.16 + 4.36i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.0980 + 0.366i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.70927161685235252166392634458, −11.31801993402788503330231055771, −10.16251207419458156377462387518, −9.053982921907077050021076608850, −8.160817116057224201840662263372, −7.49702213323867929922724178135, −6.08431315161418920363579079618, −5.44585522848869558145179753428, −4.18888201694351820522358549025, −2.62187921511745008368408118346, 1.31347903059825464755341399188, 2.90226811520123641328650619758, 3.73373233816265746266074072642, 4.94546248510286264633523335677, 6.88652062083032712304779561488, 7.48301320160365157939685343255, 9.035165037031462428385577750215, 9.876517905699346800292423618680, 10.77871587394269913833708088220, 11.53759915935103768644209254639

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