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Properties

Label 2-273-91.89-c1-0-17
Degree $2$
Conductor $273$
Sign $0.590 + 0.806i$
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  + (0.374 + 0.374i)2-s + (0.866 + 0.5i)3-s − 1.71i·4-s + (−0.545 − 2.03i)5-s + (0.137 + 0.511i)6-s + (−2.03 − 1.69i)7-s + (1.39 − 1.39i)8-s + (0.499 + 0.866i)9-s + (0.558 − 0.968i)10-s + (−0.745 − 2.78i)11-s + (0.859 − 1.48i)12-s + (2.80 + 2.27i)13-s + (−0.129 − 1.39i)14-s + (0.545 − 2.03i)15-s − 2.39·16-s − 3.29·17-s + ⋯
L(s)  = 1  + (0.264 + 0.264i)2-s + (0.499 + 0.288i)3-s − 0.859i·4-s + (−0.244 − 0.911i)5-s + (0.0559 + 0.208i)6-s + (−0.769 − 0.638i)7-s + (0.492 − 0.492i)8-s + (0.166 + 0.288i)9-s + (0.176 − 0.306i)10-s + (−0.224 − 0.839i)11-s + (0.248 − 0.429i)12-s + (0.776 + 0.629i)13-s + (−0.0345 − 0.373i)14-s + (0.140 − 0.526i)15-s − 0.598·16-s − 0.799·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.590 + 0.806i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.590 + 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36616 - 0.692787i\)
\(L(\frac12)\) \(\approx\) \(1.36616 - 0.692787i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 1.69i)T \)
13 \( 1 + (-2.80 - 2.27i)T \)
good2 \( 1 + (-0.374 - 0.374i)T + 2iT^{2} \)
5 \( 1 + (0.545 + 2.03i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.745 + 2.78i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 3.29T + 17T^{2} \)
19 \( 1 + (-6.53 - 1.75i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 7.84iT - 23T^{2} \)
29 \( 1 + (-0.677 - 1.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.38 - 1.71i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.87 + 2.87i)T - 37iT^{2} \)
41 \( 1 + (-6.61 - 1.77i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.36 + 2.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.53 - 0.947i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.87 + 6.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.01 + 2.01i)T + 59iT^{2} \)
61 \( 1 + (3.14 - 1.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.82 + 0.489i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (9.87 - 2.64i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.82 + 6.80i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.13 - 1.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.97 - 3.97i)T - 83iT^{2} \)
89 \( 1 + (-8.73 - 8.73i)T + 89iT^{2} \)
97 \( 1 + (-3.94 - 14.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.67127531327731141562571630293, −10.74838264461317671631113226616, −9.667927166587194553000186360446, −9.092805056846183760438765453196, −7.913853522248238206865477550694, −6.69534577057715874684054486955, −5.62349289951351833770843990255, −4.51490523724292993512588058099, −3.42637347494684810695708672082, −1.15811072914037836622568935525, 2.60834575809038554816647098922, 3.14665768951794374495581718769, 4.53934625853254648842982235083, 6.31175865062681714495119626961, 7.18462535783965265506816407932, 8.113136516325387232062780116969, 9.084642446959078618576981696749, 10.22523103062569125070089302593, 11.28846663291162388582858431753, 12.16085074338306728771529293559

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