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Properties

Label 2-273-39.8-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.998 + 0.0468i$
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  + (−1.08 + 1.08i)2-s + (1.14 + 1.30i)3-s − 0.353i·4-s + (−0.879 + 0.879i)5-s + (−2.65 − 0.168i)6-s + (−0.707 + 0.707i)7-s + (−1.78 − 1.78i)8-s + (−0.380 + 2.97i)9-s − 1.90i·10-s + (−0.135 − 0.135i)11-s + (0.460 − 0.404i)12-s + (−3.31 + 1.41i)13-s − 1.53i·14-s + (−2.14 − 0.136i)15-s + 4.58·16-s + 3.19·17-s + ⋯
L(s)  = 1  + (−0.767 + 0.767i)2-s + (0.660 + 0.750i)3-s − 0.176i·4-s + (−0.393 + 0.393i)5-s + (−1.08 − 0.0689i)6-s + (−0.267 + 0.267i)7-s + (−0.631 − 0.631i)8-s + (−0.126 + 0.991i)9-s − 0.603i·10-s + (−0.0407 − 0.0407i)11-s + (0.132 − 0.116i)12-s + (−0.919 + 0.393i)13-s − 0.410i·14-s + (−0.554 − 0.0353i)15-s + 1.14·16-s + 0.774·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.998 + 0.0468i$
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: \((2,\ 273,\ (\ :1/2),\ -0.998 + 0.0468i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0181482 - 0.774948i\)
\(L(\frac12)\) \(\approx\) \(0.0181482 - 0.774948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.14 - 1.30i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (3.31 - 1.41i)T \)
good2 \( 1 + (1.08 - 1.08i)T - 2iT^{2} \)
5 \( 1 + (0.879 - 0.879i)T - 5iT^{2} \)
11 \( 1 + (0.135 + 0.135i)T + 11iT^{2} \)
17 \( 1 - 3.19T + 17T^{2} \)
19 \( 1 + (0.0287 + 0.0287i)T + 19iT^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 + 2.50iT - 29T^{2} \)
31 \( 1 + (-0.0596 - 0.0596i)T + 31iT^{2} \)
37 \( 1 + (-1.80 + 1.80i)T - 37iT^{2} \)
41 \( 1 + (3.83 - 3.83i)T - 41iT^{2} \)
43 \( 1 - 4.62iT - 43T^{2} \)
47 \( 1 + (1.94 + 1.94i)T + 47iT^{2} \)
53 \( 1 - 3.75iT - 53T^{2} \)
59 \( 1 + (-10.5 - 10.5i)T + 59iT^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (-6.24 - 6.24i)T + 67iT^{2} \)
71 \( 1 + (-7.33 + 7.33i)T - 71iT^{2} \)
73 \( 1 + (-0.460 + 0.460i)T - 73iT^{2} \)
79 \( 1 - 6.30T + 79T^{2} \)
83 \( 1 + (7.87 - 7.87i)T - 83iT^{2} \)
89 \( 1 + (-8.78 - 8.78i)T + 89iT^{2} \)
97 \( 1 + (13.0 + 13.0i)T + 97iT^{2} \)
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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

−12.31313965971713207364902386554, −11.28161359349354135615377707889, −9.982449521826353111858902093235, −9.496503672639468585360598613888, −8.480398351002454210619718868644, −7.68802345781274667884475470054, −6.84829824573540060865572086693, −5.40074483370354604512943001822, −3.90290173655182775445095844813, −2.80236523943271549428825709807, 0.70203183264428279261218207446, 2.25822925537878898043008293645, 3.48996786662009889870268383948, 5.28384938468696767942323800827, 6.73731615092000852777431028552, 7.86713632086994056660095873163, 8.561638502595173371028488966743, 9.596666427459158107459991502091, 10.24745292133811028656534911066, 11.52028730013628140462958935547

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