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Properties

Label 2-273-7.4-c3-0-11
Degree $2$
Conductor $273$
Sign $-0.999 + 0.0107i$
Analytic cond. $16.1075$
Root an. cond. $4.01341$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.46 + 2.54i)2-s + (−1.5 + 2.59i)3-s + (−0.307 + 0.532i)4-s + (8.90 + 15.4i)5-s − 8.80·6-s + (−15.3 + 10.4i)7-s + 21.6·8-s + (−4.5 − 7.79i)9-s + (−26.1 + 45.2i)10-s + (−7.37 + 12.7i)11-s + (−0.922 − 1.59i)12-s − 13·13-s + (−48.9 − 23.6i)14-s − 53.4·15-s + (34.2 + 59.3i)16-s + (12.8 − 22.1i)17-s + ⋯
L(s)  = 1  + (0.518 + 0.898i)2-s + (−0.288 + 0.499i)3-s + (−0.0384 + 0.0665i)4-s + (0.796 + 1.37i)5-s − 0.599·6-s + (−0.826 + 0.562i)7-s + 0.957·8-s + (−0.166 − 0.288i)9-s + (−0.826 + 1.43i)10-s + (−0.202 + 0.350i)11-s + (−0.0221 − 0.0384i)12-s − 0.277·13-s + (−0.934 − 0.450i)14-s − 0.919·15-s + (0.535 + 0.927i)16-s + (0.182 − 0.316i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0107i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.999 + 0.0107i$
Analytic conductor: \(16.1075\)
Root analytic conductor: \(4.01341\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :3/2),\ -0.999 + 0.0107i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.192423484\)
\(L(\frac12)\) \(\approx\) \(2.192423484\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 + (15.3 - 10.4i)T \)
13 \( 1 + 13T \)
good2 \( 1 + (-1.46 - 2.54i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-8.90 - 15.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (7.37 - 12.7i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-12.8 + 22.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-46.3 - 80.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (40.5 + 70.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 75.8T + 2.43e4T^{2} \)
31 \( 1 + (31.7 - 55.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (140. + 242. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 96.3T + 6.89e4T^{2} \)
43 \( 1 - 10.5T + 7.95e4T^{2} \)
47 \( 1 + (-120. - 208. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (50.4 - 87.4i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-65.3 + 113. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-89.7 - 155. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (462. - 801. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 870.T + 3.57e5T^{2} \)
73 \( 1 + (-351. + 608. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-101. - 175. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (266. + 461. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 986.T + 9.12e5T^{2} \)
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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

−11.97367024713619436990338731991, −10.66306612034245714755001044963, −10.18508039103207032153328816745, −9.323589223432293759655161663763, −7.60057327083688856795243794090, −6.68796565525465783300691736526, −5.98794638786098212238198197888, −5.23013722160706710491930158154, −3.61257587665629741505833914976, −2.27377019989157438151020806778, 0.72152933332919884732644667902, 1.91810956550769787853106213462, 3.36682692387648757585059143916, 4.71709379236326167605465277693, 5.64355671804193258020397831410, 6.95261003585125827271803816398, 8.086927558608298618545712339484, 9.336536916795995306769037764430, 10.13893426751230252834947542432, 11.21769284079867348385533446519

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