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Properties

Label 2-273-39.8-c1-0-14
Degree $2$
Conductor $273$
Sign $0.992 - 0.123i$
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  + (1 − i)2-s + (−0.292 + 1.70i)3-s + (1.29 − 1.29i)5-s + (1.41 + 2i)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 + (1.82 + 2.58i)15-s + 4·16-s + 4·17-s + (−3.82 + 1.82i)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 + (0.577 + 0.816i)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 + (0.472 + 0.667i)15-s + 16-s + 0.970·17-s + (−0.902 + 0.430i)18-s + (−1.40 − 1.40i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.992 - 0.123i$
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: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.992 - 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90578 + 0.118389i\)
\(L(\frac12)\) \(\approx\) \(1.90578 + 0.118389i\)
\(L(\frac{3}{2})\) 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

−11.68731077388293503606539996111, −11.28252607406368962480612661512, −10.13321682874859008840486012136, −9.290821600593297536921905349395, −8.359353342319508761321387458560, −6.80025269453675098597481583672, −5.26614995240576098841311492731, −4.60242689833863846093487179547, −3.63487589580332348181541940916, −2.06775714843674350674621483690, 1.61371997258903642255492442965, 3.34377727841273432338380767964, 5.23239027688109253442316178727, 5.90741961748669634049818250843, 6.69955002229246868456957204888, 7.64640171933969498915770182426, 8.710882544546766186518915557275, 10.28732167854631408779402749236, 10.85164659250631372236380009273, 12.39211285955535432475852711662

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