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Properties

Label 2-273-7.2-c3-0-27
Degree $2$
Conductor $273$
Sign $0.803 - 0.595i$
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  + (−1.79 + 3.11i)2-s + (−1.5 − 2.59i)3-s + (−2.47 − 4.28i)4-s + (−0.896 + 1.55i)5-s + 10.7·6-s + (18.4 − 0.950i)7-s − 10.9·8-s + (−4.5 + 7.79i)9-s + (−3.22 − 5.58i)10-s + (−1.86 − 3.23i)11-s + (−7.41 + 12.8i)12-s − 13·13-s + (−30.3 + 59.3i)14-s + 5.38·15-s + (39.5 − 68.5i)16-s + (−28.5 − 49.4i)17-s + ⋯
L(s)  = 1  + (−0.636 + 1.10i)2-s + (−0.288 − 0.499i)3-s + (−0.309 − 0.535i)4-s + (−0.0802 + 0.138i)5-s + 0.734·6-s + (0.998 − 0.0513i)7-s − 0.485·8-s + (−0.166 + 0.288i)9-s + (−0.102 − 0.176i)10-s + (−0.0512 − 0.0887i)11-s + (−0.178 + 0.309i)12-s − 0.277·13-s + (−0.578 + 1.13i)14-s + 0.0926·15-s + (0.618 − 1.07i)16-s + (−0.407 − 0.705i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.803 - 0.595i$
Analytic conductor: \(16.1075\)
Root analytic conductor: \(4.01341\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :3/2),\ 0.803 - 0.595i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.159048399\)
\(L(\frac12)\) \(\approx\) \(1.159048399\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 + 0.950i)T \)
13 \( 1 + 13T \)
good2 \( 1 + (1.79 - 3.11i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (0.896 - 1.55i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (1.86 + 3.23i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (28.5 + 49.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-41.8 + 72.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-35.7 + 61.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 66.4T + 2.43e4T^{2} \)
31 \( 1 + (-67.0 - 116. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-112. + 194. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 232.T + 6.89e4T^{2} \)
43 \( 1 - 524.T + 7.95e4T^{2} \)
47 \( 1 + (-76.8 + 133. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-231. - 401. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-213. - 370. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (129. - 224. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-65.8 - 114. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 551.T + 3.57e5T^{2} \)
73 \( 1 + (111. + 193. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-235. + 407. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 360.T + 5.71e5T^{2} \)
89 \( 1 + (-354. + 613. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 562.T + 9.12e5T^{2} \)
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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

−11.53451937286643934645195776018, −10.65915334218935965651298527533, −9.215996574962946407996223845671, −8.512734514716970832696470509844, −7.39857893025354563515846169312, −7.02769788091814354489491485085, −5.73258529517903271208602781765, −4.75615463238603816950215642914, −2.68562614112447221814925451327, −0.76672972816991804587646911068, 0.998369375918931177820236516061, 2.32921773197100663825721697200, 3.82377742425709016171828819845, 5.02729836551228304951942489616, 6.23541029955581734749235761919, 7.893803922185535250947617531453, 8.718816505743106320180475756978, 9.748002692790984002701463986493, 10.46843046144854626138259856080, 11.27695028117184206334439228253

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