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Properties

Label 2-273-91.34-c1-0-9
Degree $2$
Conductor $273$
Sign $0.982 + 0.187i$
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  + i·3-s − 2i·4-s + (0.581 − 0.581i)5-s + (0.581 + 2.58i)7-s − 9-s + (4.16 − 4.16i)11-s + 2·12-s + (3.58 + 0.418i)13-s + (0.581 + 0.581i)15-s − 4·16-s + 1.16·17-s + (0.418 − 0.418i)19-s + (−1.16 − 1.16i)20-s + (−2.58 + 0.581i)21-s − 4.16i·23-s + ⋯
L(s)  = 1  + 0.577i·3-s − i·4-s + (0.259 − 0.259i)5-s + (0.219 + 0.975i)7-s − 0.333·9-s + (1.25 − 1.25i)11-s + 0.577·12-s + (0.993 + 0.116i)13-s + (0.150 + 0.150i)15-s − 16-s + 0.281·17-s + (0.0960 − 0.0960i)19-s + (−0.259 − 0.259i)20-s + (−0.563 + 0.126i)21-s − 0.867i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.982 + 0.187i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.982 + 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41186 - 0.133805i\)
\(L(\frac12)\) \(\approx\) \(1.41186 - 0.133805i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.581 - 2.58i)T \)
13 \( 1 + (-3.58 - 0.418i)T \)
good2 \( 1 + 2iT^{2} \)
5 \( 1 + (-0.581 + 0.581i)T - 5iT^{2} \)
11 \( 1 + (-4.16 + 4.16i)T - 11iT^{2} \)
17 \( 1 - 1.16T + 17T^{2} \)
19 \( 1 + (-0.418 + 0.418i)T - 19iT^{2} \)
23 \( 1 + 4.16iT - 23T^{2} \)
29 \( 1 - 1.83T + 29T^{2} \)
31 \( 1 + (5.58 - 5.58i)T - 31iT^{2} \)
37 \( 1 + (-4.32 + 4.32i)T - 37iT^{2} \)
41 \( 1 + (7.16 - 7.16i)T - 41iT^{2} \)
43 \( 1 - 5.32iT - 43T^{2} \)
47 \( 1 + (6.58 + 6.58i)T + 47iT^{2} \)
53 \( 1 + 4.16T + 53T^{2} \)
59 \( 1 + (8.32 + 8.32i)T + 59iT^{2} \)
61 \( 1 - 3.16iT - 61T^{2} \)
67 \( 1 + (6.32 + 6.32i)T + 67iT^{2} \)
71 \( 1 + (-6 - 6i)T + 71iT^{2} \)
73 \( 1 + (-5.58 - 5.58i)T + 73iT^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + (6.58 - 6.58i)T - 83iT^{2} \)
89 \( 1 + (5.41 + 5.41i)T + 89iT^{2} \)
97 \( 1 + (0.418 - 0.418i)T - 97iT^{2} \)
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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

−11.41774242428083303342170067553, −11.15234368364671544837891644650, −9.868549108521000633149937470469, −8.998306864406598237574962590345, −8.530059442698811856848207744950, −6.46344629272368819701451836386, −5.83391123504976754757562748370, −4.85540221144306396035516576361, −3.35960043393467267876393293161, −1.45362398774215730435941109604, 1.69721431646154716020872007460, 3.46691423296353433533872079049, 4.42305808868078945862088202919, 6.26632381226452616925485420693, 7.13772476495172290886558007442, 7.82774656883646789471256226219, 8.959036167365525532083839469251, 10.02427229453914790958913810543, 11.23668243389477309302486688417, 11.96050400910198618626085644289

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