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Properties

Label 2-273-39.8-c1-0-9
Degree $2$
Conductor $273$
Sign $-0.326 - 0.945i$
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  + (−1.65 + 1.65i)2-s + (0.818 + 1.52i)3-s − 3.46i·4-s + (1.96 − 1.96i)5-s + (−3.87 − 1.17i)6-s + (0.707 − 0.707i)7-s + (2.42 + 2.42i)8-s + (−1.66 + 2.49i)9-s + 6.48i·10-s + (2.17 + 2.17i)11-s + (5.28 − 2.83i)12-s + (2.86 + 2.18i)13-s + 2.33i·14-s + (4.60 + 1.39i)15-s − 1.07·16-s − 3.32·17-s + ⋯
L(s)  = 1  + (−1.16 + 1.16i)2-s + (0.472 + 0.881i)3-s − 1.73i·4-s + (0.877 − 0.877i)5-s + (−1.58 − 0.478i)6-s + (0.267 − 0.267i)7-s + (0.855 + 0.855i)8-s + (−0.553 + 0.832i)9-s + 2.05i·10-s + (0.655 + 0.655i)11-s + (1.52 − 0.818i)12-s + (0.795 + 0.605i)13-s + 0.624i·14-s + (1.18 + 0.359i)15-s − 0.268·16-s − 0.807·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.326 - 0.945i$
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: \((2,\ 273,\ (\ :1/2),\ -0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567915 + 0.797363i\)
\(L(\frac12)\) \(\approx\) \(0.567915 + 0.797363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.818 - 1.52i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-2.86 - 2.18i)T \)
good2 \( 1 + (1.65 - 1.65i)T - 2iT^{2} \)
5 \( 1 + (-1.96 + 1.96i)T - 5iT^{2} \)
11 \( 1 + (-2.17 - 2.17i)T + 11iT^{2} \)
17 \( 1 + 3.32T + 17T^{2} \)
19 \( 1 + (-2.91 - 2.91i)T + 19iT^{2} \)
23 \( 1 + 0.190T + 23T^{2} \)
29 \( 1 + 7.09iT - 29T^{2} \)
31 \( 1 + (-1.90 - 1.90i)T + 31iT^{2} \)
37 \( 1 + (6.97 - 6.97i)T - 37iT^{2} \)
41 \( 1 + (-6.16 + 6.16i)T - 41iT^{2} \)
43 \( 1 + 12.4iT - 43T^{2} \)
47 \( 1 + (-2.28 - 2.28i)T + 47iT^{2} \)
53 \( 1 - 5.76iT - 53T^{2} \)
59 \( 1 + (-2.95 - 2.95i)T + 59iT^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (4.16 + 4.16i)T + 67iT^{2} \)
71 \( 1 + (-1.60 + 1.60i)T - 71iT^{2} \)
73 \( 1 + (-0.722 + 0.722i)T - 73iT^{2} \)
79 \( 1 - 7.14T + 79T^{2} \)
83 \( 1 + (-3.15 + 3.15i)T - 83iT^{2} \)
89 \( 1 + (5.99 + 5.99i)T + 89iT^{2} \)
97 \( 1 + (7.27 + 7.27i)T + 97iT^{2} \)
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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

−12.05735995407687910275459837626, −10.65649761557392156830318808629, −9.843342308561231801275429868630, −9.093776416758169166879049714906, −8.676453995008423903706044030006, −7.55210010131375368324851412530, −6.31913675997780710780402005475, −5.31575226963250324045529607175, −4.13671359361334021869728965629, −1.65452918853732522295554784798, 1.26694013496578792367123499608, 2.49771070354464264341594176313, 3.34866353076392944291641276507, 5.90473695246011202140194540913, 6.89959048357303650680728138081, 8.107246484441743002150501015964, 8.929418619204200375585046066978, 9.595718320472319639401338380611, 10.88485466153961011374642624210, 11.23574754766207536714149170586

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