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Properties

Label 2-273-39.8-c1-0-22
Degree $2$
Conductor $273$
Sign $-0.894 + 0.447i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−0.292 − 1.70i)3-s + (−1.29 + 1.29i)5-s + (2 + 1.41i)6-s + (0.707 − 0.707i)7-s + (−2 − 2i)8-s + (−2.82 + i)9-s − 2.58i·10-s + (−1.41 − 1.41i)11-s + (−0.707 + 3.53i)13-s + 1.41i·14-s + (2.58 + 1.82i)15-s + 4·16-s − 4·17-s + (1.82 − 3.82i)18-s + (−6.12 − 6.12i)19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.169 − 0.985i)3-s + (−0.578 + 0.578i)5-s + (0.816 + 0.577i)6-s + (0.267 − 0.267i)7-s + (−0.707 − 0.707i)8-s + (−0.942 + 0.333i)9-s − 0.817i·10-s + (−0.426 − 0.426i)11-s + (−0.196 + 0.980i)13-s + 0.377i·14-s + (0.667 + 0.472i)15-s + 16-s − 0.970·17-s + (0.430 − 0.902i)18-s + (−1.40 − 1.40i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.292 + 1.70i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.707 - 3.53i)T \)
good2 \( 1 + (1 - i)T - 2iT^{2} \)
5 \( 1 + (1.29 - 1.29i)T - 5iT^{2} \)
11 \( 1 + (1.41 + 1.41i)T + 11iT^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + (6.12 + 6.12i)T + 19iT^{2} \)
23 \( 1 + 3.82T + 23T^{2} \)
29 \( 1 - 4.65iT - 29T^{2} \)
31 \( 1 + (6.94 + 6.94i)T + 31iT^{2} \)
37 \( 1 + (3.58 - 3.58i)T - 37iT^{2} \)
41 \( 1 + (-8.24 + 8.24i)T - 41iT^{2} \)
43 \( 1 + 4.65iT - 43T^{2} \)
47 \( 1 + (-6.12 - 6.12i)T + 47iT^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + (-1.41 - 1.41i)T + 59iT^{2} \)
61 \( 1 - 5.89T + 61T^{2} \)
67 \( 1 + (-5.07 - 5.07i)T + 67iT^{2} \)
71 \( 1 + (2.75 - 2.75i)T - 71iT^{2} \)
73 \( 1 + (4.12 - 4.12i)T - 73iT^{2} \)
79 \( 1 - 0.171T + 79T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - 83iT^{2} \)
89 \( 1 + (10.1 + 10.1i)T + 89iT^{2} \)
97 \( 1 + (8.70 + 8.70i)T + 97iT^{2} \)
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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

−11.35309477758938677140877084214, −10.82821273194896228462212459858, −9.082009703823424862762389194386, −8.483348024469992215282166338049, −7.27051829409949760496563209021, −7.06615897152807525921747390704, −5.93105854588474171452825919770, −4.10688183925757030996443034502, −2.44398454524324804949917424868, 0, 2.27365278627532826967251926893, 3.92180989062329451106168951247, 5.05720555293111183915009797970, 6.05099925858386684562592065532, 8.097983645952740801896972106241, 8.581873106039718668217667362342, 9.667613855411056649219594034872, 10.43762414654910847956480708088, 11.04931370916904285567906799812

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