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Properties

Label 2-273-21.20-c1-0-31
Degree $2$
Conductor $273$
Sign $-0.225 - 0.974i$
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  − 2.62i·2-s + (0.518 − 1.65i)3-s − 4.91·4-s − 2.48·5-s + (−4.34 − 1.36i)6-s + (1.33 + 2.28i)7-s + 7.67i·8-s + (−2.46 − 1.71i)9-s + 6.54i·10-s − 5.20i·11-s + (−2.54 + 8.12i)12-s + i·13-s + (6.00 − 3.52i)14-s + (−1.28 + 4.11i)15-s + 10.3·16-s − 2.44·17-s + ⋯
L(s)  = 1  − 1.85i·2-s + (0.299 − 0.954i)3-s − 2.45·4-s − 1.11·5-s + (−1.77 − 0.556i)6-s + (0.506 + 0.862i)7-s + 2.71i·8-s + (−0.820 − 0.570i)9-s + 2.07i·10-s − 1.56i·11-s + (−0.735 + 2.34i)12-s + 0.277i·13-s + (1.60 − 0.941i)14-s + (−0.333 + 1.06i)15-s + 2.58·16-s − 0.592·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.225 - 0.974i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.225 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464121 + 0.583623i\)
\(L(\frac12)\) \(\approx\) \(0.464121 + 0.583623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.518 + 1.65i)T \)
7 \( 1 + (-1.33 - 2.28i)T \)
13 \( 1 - iT \)
good2 \( 1 + 2.62iT - 2T^{2} \)
5 \( 1 + 2.48T + 5T^{2} \)
11 \( 1 + 5.20iT - 11T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 + 5.95iT - 23T^{2} \)
29 \( 1 - 2.80iT - 29T^{2} \)
31 \( 1 + 6.81iT - 31T^{2} \)
37 \( 1 - 3.17T + 37T^{2} \)
41 \( 1 + 0.133T + 41T^{2} \)
43 \( 1 - 6.61T + 43T^{2} \)
47 \( 1 - 4.15T + 47T^{2} \)
53 \( 1 - 7.32iT - 53T^{2} \)
59 \( 1 - 0.723T + 59T^{2} \)
61 \( 1 + 8.23iT - 61T^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 + 7.17iT - 71T^{2} \)
73 \( 1 + 16.4iT - 73T^{2} \)
79 \( 1 + 0.0675T + 79T^{2} \)
83 \( 1 - 6.38T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 + 16.7iT - 97T^{2} \)
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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

−11.31107522725811380363744026861, −10.94037357338912239997087838872, −9.119766425264766135951318491596, −8.640285023778598964068083776183, −7.81641276368363184362541454171, −6.01049969449945589883036517743, −4.48272116109051285097507619345, −3.25849931134787116807465232605, −2.28679992986246177901721730736, −0.56795315912200284047523753655, 3.92027185170160911673457955854, 4.42896156860497995579925692023, 5.41643386655647174145663932989, 7.02886223172919545444479014657, 7.65824129523511978681622566178, 8.362211238497765960385795437236, 9.468330662378437235915640125586, 10.31584939883370876194001707377, 11.58535066634171648003749818951, 12.91972037383554135392539106515

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