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Properties

Label 2-273-273.272-c1-0-15
Degree $2$
Conductor $273$
Sign $0.271 - 0.962i$
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  + 1.10·2-s + (0.669 + 1.59i)3-s − 0.773·4-s + 1.62i·5-s + (0.741 + 1.76i)6-s + (2.53 − 0.762i)7-s − 3.07·8-s + (−2.10 + 2.13i)9-s + 1.80i·10-s + 2.37·11-s + (−0.517 − 1.23i)12-s + (2.05 + 2.96i)13-s + (2.80 − 0.844i)14-s + (−2.59 + 1.08i)15-s − 1.85·16-s − 5.91·17-s + ⋯
L(s)  = 1  + 0.783·2-s + (0.386 + 0.922i)3-s − 0.386·4-s + 0.727i·5-s + (0.302 + 0.722i)6-s + (0.957 − 0.288i)7-s − 1.08·8-s + (−0.701 + 0.712i)9-s + 0.569i·10-s + 0.715·11-s + (−0.149 − 0.356i)12-s + (0.570 + 0.821i)13-s + (0.749 − 0.225i)14-s + (−0.670 + 0.281i)15-s − 0.463·16-s − 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.271 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49897 + 1.13477i\)
\(L(\frac12)\) \(\approx\) \(1.49897 + 1.13477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 - 1.59i)T \)
7 \( 1 + (-2.53 + 0.762i)T \)
13 \( 1 + (-2.05 - 2.96i)T \)
good2 \( 1 - 1.10T + 2T^{2} \)
5 \( 1 - 1.62iT - 5T^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
17 \( 1 + 5.91T + 17T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 + 7.30iT - 23T^{2} \)
29 \( 1 + 1.65iT - 29T^{2} \)
31 \( 1 - 3.64T + 31T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 + 2.27iT - 41T^{2} \)
43 \( 1 - 2.85T + 43T^{2} \)
47 \( 1 + 9.20iT - 47T^{2} \)
53 \( 1 - 12.3iT - 53T^{2} \)
59 \( 1 - 9.86iT - 59T^{2} \)
61 \( 1 - 2.47iT - 61T^{2} \)
67 \( 1 - 5.91iT - 67T^{2} \)
71 \( 1 + 8.43T + 71T^{2} \)
73 \( 1 + 7.77T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 5.44iT - 83T^{2} \)
89 \( 1 + 10.4iT - 89T^{2} \)
97 \( 1 + 4.17T + 97T^{2} \)
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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

−11.96512908607264489683748015524, −11.14260884634051554942110152471, −10.40072043469215830645710352396, −9.015580281160272611765077103730, −8.647339689770784538135811027749, −7.03116432642668460791558742705, −5.83295470112756470624198513155, −4.43764456940620397100718864440, −4.10542306148179770482639659174, −2.58668253072765851675599006846, 1.34458688415895151295402685976, 3.14837406237106339031033677874, 4.52663748220722164904053575334, 5.49738851363529275656260720100, 6.56172366183611926093916167079, 8.019920222419816131726860424020, 8.668469050131240789087233334694, 9.437847086293542990069955827939, 11.31234547886741217407777929732, 11.90120881168801098339970884785

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