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Properties

Label 2-273-91.83-c1-0-12
Degree $2$
Conductor $273$
Sign $0.370 + 0.928i$
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  + (−0.786 + 0.786i)2-s + i·3-s + 0.762i·4-s + (−2.85 − 2.85i)5-s + (−0.786 − 0.786i)6-s + (1.90 − 1.83i)7-s + (−2.17 − 2.17i)8-s − 9-s + 4.48·10-s + (−1.37 − 1.37i)11-s − 0.762·12-s + (−2.79 − 2.27i)13-s + (−0.0482 + 2.94i)14-s + (2.85 − 2.85i)15-s + 1.89·16-s + 3.67·17-s + ⋯
L(s)  = 1  + (−0.556 + 0.556i)2-s + 0.577i·3-s + 0.381i·4-s + (−1.27 − 1.27i)5-s + (−0.321 − 0.321i)6-s + (0.718 − 0.695i)7-s + (−0.768 − 0.768i)8-s − 0.333·9-s + 1.41·10-s + (−0.413 − 0.413i)11-s − 0.220·12-s + (−0.775 − 0.631i)13-s + (−0.0128 + 0.786i)14-s + (0.736 − 0.736i)15-s + 0.473·16-s + 0.892·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.370 + 0.928i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.370 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378276 - 0.256255i\)
\(L(\frac12)\) \(\approx\) \(0.378276 - 0.256255i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-1.90 + 1.83i)T \)
13 \( 1 + (2.79 + 2.27i)T \)
good2 \( 1 + (0.786 - 0.786i)T - 2iT^{2} \)
5 \( 1 + (2.85 + 2.85i)T + 5iT^{2} \)
11 \( 1 + (1.37 + 1.37i)T + 11iT^{2} \)
17 \( 1 - 3.67T + 17T^{2} \)
19 \( 1 + (1.10 + 1.10i)T + 19iT^{2} \)
23 \( 1 + 0.653iT - 23T^{2} \)
29 \( 1 + 1.78T + 29T^{2} \)
31 \( 1 + (5.70 + 5.70i)T + 31iT^{2} \)
37 \( 1 + (-3.46 - 3.46i)T + 37iT^{2} \)
41 \( 1 + (2.67 + 2.67i)T + 41iT^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + (3.84 - 3.84i)T - 47iT^{2} \)
53 \( 1 + 0.919T + 53T^{2} \)
59 \( 1 + (8.69 - 8.69i)T - 59iT^{2} \)
61 \( 1 + 5.92iT - 61T^{2} \)
67 \( 1 + (8.44 - 8.44i)T - 67iT^{2} \)
71 \( 1 + (-8.55 + 8.55i)T - 71iT^{2} \)
73 \( 1 + (6.74 - 6.74i)T - 73iT^{2} \)
79 \( 1 - 9.87T + 79T^{2} \)
83 \( 1 + (-8.71 - 8.71i)T + 83iT^{2} \)
89 \( 1 + (-4.94 + 4.94i)T - 89iT^{2} \)
97 \( 1 + (-0.857 - 0.857i)T + 97iT^{2} \)
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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

−11.78651422976152869519122493916, −10.76981908887141817985758159461, −9.547182945312203207508909443459, −8.580607343127449437639519176198, −7.895853043874680782791535407560, −7.39169876516774199393792204800, −5.44658684901426662514504654255, −4.42988273915467679594317405703, −3.47282291209588071685169537494, −0.41193493775835174196050110815, 1.97379855976269592823587300174, 3.13535080102031077376728734205, 4.92020345948450072312069805793, 6.28132690136687945638029486187, 7.45787194032663677632594460927, 8.075298456064072531688213796518, 9.307712387416973193890424747658, 10.43154334233703863901362574578, 11.20735871504640941883973916213, 11.81475068272070094905508059874

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