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Properties

Label 2-273-39.5-c1-0-4
Degree $2$
Conductor $273$
Sign $-0.688 - 0.724i$
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  + (1.89 + 1.89i)2-s + (−1.56 − 0.731i)3-s + 5.20i·4-s + (0.367 + 0.367i)5-s + (−1.59 − 4.36i)6-s + (0.707 + 0.707i)7-s + (−6.09 + 6.09i)8-s + (1.92 + 2.29i)9-s + 1.39i·10-s + (−0.446 + 0.446i)11-s + (3.81 − 8.17i)12-s + (−3.58 + 0.419i)13-s + 2.68i·14-s + (−0.307 − 0.845i)15-s − 12.7·16-s + 2.33·17-s + ⋯
L(s)  = 1  + (1.34 + 1.34i)2-s + (−0.906 − 0.422i)3-s + 2.60i·4-s + (0.164 + 0.164i)5-s + (−0.649 − 1.78i)6-s + (0.267 + 0.267i)7-s + (−2.15 + 2.15i)8-s + (0.643 + 0.765i)9-s + 0.441i·10-s + (−0.134 + 0.134i)11-s + (1.10 − 2.36i)12-s + (−0.993 + 0.116i)13-s + 0.717i·14-s + (−0.0795 − 0.218i)15-s − 3.17·16-s + 0.566·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.688 - 0.724i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.688 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.747952 + 1.74260i\)
\(L(\frac12)\) \(\approx\) \(0.747952 + 1.74260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56 + 0.731i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (3.58 - 0.419i)T \)
good2 \( 1 + (-1.89 - 1.89i)T + 2iT^{2} \)
5 \( 1 + (-0.367 - 0.367i)T + 5iT^{2} \)
11 \( 1 + (0.446 - 0.446i)T - 11iT^{2} \)
17 \( 1 - 2.33T + 17T^{2} \)
19 \( 1 + (-3.79 + 3.79i)T - 19iT^{2} \)
23 \( 1 - 6.50T + 23T^{2} \)
29 \( 1 - 2.56iT - 29T^{2} \)
31 \( 1 + (0.353 - 0.353i)T - 31iT^{2} \)
37 \( 1 + (0.259 + 0.259i)T + 37iT^{2} \)
41 \( 1 + (-7.20 - 7.20i)T + 41iT^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (-4.63 + 4.63i)T - 47iT^{2} \)
53 \( 1 - 8.90iT - 53T^{2} \)
59 \( 1 + (4.44 - 4.44i)T - 59iT^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (-7.41 + 7.41i)T - 67iT^{2} \)
71 \( 1 + (10.2 + 10.2i)T + 71iT^{2} \)
73 \( 1 + (7.54 + 7.54i)T + 73iT^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 + (7.31 + 7.31i)T + 83iT^{2} \)
89 \( 1 + (2.01 - 2.01i)T - 89iT^{2} \)
97 \( 1 + (3.54 - 3.54i)T - 97iT^{2} \)
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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

−12.31816573778302800489899849422, −11.92150034731186226868622666885, −10.70571396345196781935079825529, −9.097429576059577703207577840145, −7.64521648891180779123721444047, −7.17493903923042973681643664567, −6.15384469151342578483989919619, −5.21840607050926037106587534392, −4.60021363625323138590205233194, −2.80691714948966861473635543177, 1.22675454418532878345635142278, 3.05700919257521795504363123658, 4.26183032752375577202477489490, 5.20072328269437925226015332266, 5.81177696269984924603209246782, 7.26408115139756017661742307086, 9.443383612176846951877215862579, 10.02445318264321692651370526986, 10.97305192242461454043794505536, 11.55728248560765579340165495595

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