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Properties

Label 2-273-13.12-c1-0-4
Degree $2$
Conductor $273$
Sign $-0.798 - 0.601i$
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  + 2.17i·2-s + 3-s − 2.70·4-s − 0.630i·5-s + 2.17i·6-s + i·7-s − 1.53i·8-s + 9-s + 1.36·10-s + 5.70i·11-s − 2.70·12-s + (−2.17 + 2.87i)13-s − 2.17·14-s − 0.630i·15-s − 2.07·16-s + 1.07·17-s + ⋯
L(s)  = 1  + 1.53i·2-s + 0.577·3-s − 1.35·4-s − 0.282i·5-s + 0.885i·6-s + 0.377i·7-s − 0.544i·8-s + 0.333·9-s + 0.432·10-s + 1.72i·11-s − 0.782·12-s + (−0.601 + 0.798i)13-s − 0.579·14-s − 0.162i·15-s − 0.519·16-s + 0.261·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.798 - 0.601i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.798 - 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.451937 + 1.35053i\)
\(L(\frac12)\) \(\approx\) \(0.451937 + 1.35053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - iT \)
13 \( 1 + (2.17 - 2.87i)T \)
good2 \( 1 - 2.17iT - 2T^{2} \)
5 \( 1 + 0.630iT - 5T^{2} \)
11 \( 1 - 5.70iT - 11T^{2} \)
17 \( 1 - 1.07T + 17T^{2} \)
19 \( 1 + 7.41iT - 19T^{2} \)
23 \( 1 - 5.41T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + 6.15iT - 31T^{2} \)
37 \( 1 + 3.41iT - 37T^{2} \)
41 \( 1 + 1.21iT - 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 + 6.04iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 1.36iT - 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 + 0.581iT - 67T^{2} \)
71 \( 1 - 4.81iT - 71T^{2} \)
73 \( 1 - 3.81iT - 73T^{2} \)
79 \( 1 + 1.65T + 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + 6.20iT - 89T^{2} \)
97 \( 1 - 1.07iT - 97T^{2} \)
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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

−12.60025084524826471231356665192, −11.46837167978456605611003492691, −9.839564182423097390096652620144, −9.130046766844902550691042705760, −8.336436245641758645490335419890, −7.05065626201197100117786875001, −6.87077313329531232272053978931, −5.07896967136810093548979957172, −4.55756201184173832506471217065, −2.40523797121673712154553993532, 1.16562027456661869061557140713, 3.00785803725823986099478664975, 3.41077632709286330860510459863, 4.98543438906148982988847618257, 6.56506880454803847528069253209, 8.034362462366570949678226393236, 8.811975164797798970238473440675, 10.08657142583933243525912579529, 10.50058292325122204817204286754, 11.43432955357083461865826726520

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