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Properties

Label 2-273-91.30-c1-0-15
Degree $2$
Conductor $273$
Sign $0.625 + 0.780i$
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  + (1.92 − 1.11i)2-s + 3-s + (1.46 − 2.54i)4-s + (0.701 + 0.404i)5-s + (1.92 − 1.11i)6-s + (−2.09 + 1.61i)7-s − 2.08i·8-s + 9-s + 1.80·10-s − 3.07i·11-s + (1.46 − 2.54i)12-s + (−3.01 − 1.97i)13-s + (−2.24 + 5.43i)14-s + (0.701 + 0.404i)15-s + (0.618 + 1.07i)16-s + (−1.39 + 2.42i)17-s + ⋯
L(s)  = 1  + (1.36 − 0.785i)2-s + 0.577·3-s + (0.734 − 1.27i)4-s + (0.313 + 0.181i)5-s + (0.785 − 0.453i)6-s + (−0.792 + 0.610i)7-s − 0.738i·8-s + 0.333·9-s + 0.569·10-s − 0.927i·11-s + (0.424 − 0.734i)12-s + (−0.836 − 0.548i)13-s + (−0.599 + 1.45i)14-s + (0.181 + 0.104i)15-s + (0.154 + 0.267i)16-s + (−0.339 + 0.588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.625 + 0.780i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52805 - 1.21326i\)
\(L(\frac12)\) \(\approx\) \(2.52805 - 1.21326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (2.09 - 1.61i)T \)
13 \( 1 + (3.01 + 1.97i)T \)
good2 \( 1 + (-1.92 + 1.11i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.701 - 0.404i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.07iT - 11T^{2} \)
17 \( 1 + (1.39 - 2.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 4.31iT - 19T^{2} \)
23 \( 1 + (2.47 + 4.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.84 - 4.93i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.93 - 1.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.42 + 4.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.48 - 4.90i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.85 + 4.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.31 - 3.06i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.83 - 4.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.27 + 2.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.03T + 61T^{2} \)
67 \( 1 + 9.92iT - 67T^{2} \)
71 \( 1 + (7.84 - 4.52i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.75 + 1.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.13 + 1.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.52iT - 83T^{2} \)
89 \( 1 + (-2.87 + 1.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.0 + 6.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.23475718528190030610344710243, −10.91143743630564348841639535383, −10.18224654064074972624864293042, −9.072704094622132041867830429990, −7.917895031399119009096123877111, −6.26435838595210286867632500486, −5.63635565769084611734836699197, −4.17224473708827762751107753288, −3.12212258888688375784651965696, −2.24029059950891317268629658653, 2.53217799946920522199119659060, 3.92744425329664366665875039821, 4.73547974248722866076809041447, 5.98320141395695467808392042167, 7.14573454450916461136186874325, 7.50508896395442535030713656849, 9.402006949958039951833141005022, 9.773458075754042451856487801618, 11.47054036988522493974097046903, 12.48821847064937052853685149272

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