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Properties

Label 2-273-91.30-c1-0-6
Degree $2$
Conductor $273$
Sign $0.998 - 0.0564i$
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  + (−2.17 + 1.25i)2-s − 3-s + (2.14 − 3.71i)4-s + (2.97 + 1.71i)5-s + (2.17 − 1.25i)6-s + (−2.31 − 1.28i)7-s + 5.75i·8-s + 9-s − 8.60·10-s − 5.02i·11-s + (−2.14 + 3.71i)12-s + (0.135 − 3.60i)13-s + (6.63 − 0.118i)14-s + (−2.97 − 1.71i)15-s + (−2.92 − 5.07i)16-s + (−0.0996 + 0.172i)17-s + ⋯
L(s)  = 1  + (−1.53 + 0.887i)2-s − 0.577·3-s + (1.07 − 1.85i)4-s + (1.32 + 0.766i)5-s + (0.887 − 0.512i)6-s + (−0.874 − 0.484i)7-s + 2.03i·8-s + 0.333·9-s − 2.72·10-s − 1.51i·11-s + (−0.619 + 1.07i)12-s + (0.0375 − 0.999i)13-s + (1.77 − 0.0317i)14-s + (−0.766 − 0.442i)15-s + (−0.732 − 1.26i)16-s + (−0.0241 + 0.0418i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.998 - 0.0564i$
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: \((2,\ 273,\ (\ :1/2),\ 0.998 - 0.0564i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554074 + 0.0156540i\)
\(L(\frac12)\) \(\approx\) \(0.554074 + 0.0156540i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.31 + 1.28i)T \)
13 \( 1 + (-0.135 + 3.60i)T \)
good2 \( 1 + (2.17 - 1.25i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.97 - 1.71i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 5.02iT - 11T^{2} \)
17 \( 1 + (0.0996 - 0.172i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 4.61iT - 19T^{2} \)
23 \( 1 + (-3.76 - 6.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.58 + 2.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.65 + 2.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.89 + 4.55i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.67 - 2.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.27 - 2.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.19 + 3.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.215 + 0.373i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.725 + 0.419i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.66T + 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 + (4.95 - 2.86i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.73 - 1.57i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.25 + 3.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + (2.03 - 1.17i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.02 - 2.89i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.14520178018287836555659870951, −10.75146384960674477375448865884, −9.772606530382460225285428412861, −9.319719601033042713870820342302, −7.972304137193726456123356903598, −6.90879474271072141000842369827, −6.14727856546959762384076945295, −5.62240848606196308941372352153, −2.91964808654738574274650295649, −0.817529920739538024672302034950, 1.44491723954919343814276426484, 2.51277089221931142921252980508, 4.62964750234377898012754778079, 6.14681252220987699295042596837, 7.05945075123032739857982158663, 8.561141488807066462470838786174, 9.400391721103480058244894056567, 9.843121313137574910774143276548, 10.55185590495640291188012745237, 11.90533076704826005213250597034

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