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Properties

Label 2-273-7.2-c1-0-4
Degree $2$
Conductor $273$
Sign $0.386 - 0.922i$
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  + (−0.400 + 0.694i)2-s + (−0.5 − 0.866i)3-s + (0.678 + 1.17i)4-s + (−0.346 + 0.599i)5-s + 0.801·6-s + (2.20 − 1.46i)7-s − 2.69·8-s + (−0.499 + 0.866i)9-s + (−0.277 − 0.480i)10-s + (2.82 + 4.89i)11-s + (0.678 − 1.17i)12-s + 13-s + (0.134 + 2.11i)14-s + 0.692·15-s + (−0.277 + 0.480i)16-s + (0.266 + 0.461i)17-s + ⋯
L(s)  = 1  + (−0.283 + 0.491i)2-s + (−0.288 − 0.499i)3-s + (0.339 + 0.587i)4-s + (−0.154 + 0.268i)5-s + 0.327·6-s + (0.832 − 0.553i)7-s − 0.951·8-s + (−0.166 + 0.288i)9-s + (−0.0877 − 0.151i)10-s + (0.852 + 1.47i)11-s + (0.195 − 0.339i)12-s + 0.277·13-s + (0.0359 + 0.565i)14-s + 0.178·15-s + (−0.0693 + 0.120i)16-s + (0.0646 + 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946862 + 0.629836i\)
\(L(\frac12)\) \(\approx\) \(0.946862 + 0.629836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.20 + 1.46i)T \)
13 \( 1 - T \)
good2 \( 1 + (0.400 - 0.694i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.346 - 0.599i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.82 - 4.89i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.266 - 0.461i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.698 + 1.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.67 - 2.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.80T + 29T^{2} \)
31 \( 1 + (-2.59 - 4.50i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.06 + 7.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.80T + 41T^{2} \)
43 \( 1 + 8.96T + 43T^{2} \)
47 \( 1 + (-5.98 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.11 + 1.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.25 + 5.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.67 + 8.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.12 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.97T + 71T^{2} \)
73 \( 1 + (-0.818 - 1.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.20 - 7.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.44T + 83T^{2} \)
89 \( 1 + (-7.49 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.0827T + 97T^{2} \)
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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

−11.93205456170248358338390313193, −11.41065729568701472909974109925, −10.23613147262923977701974964008, −8.968625444825070482807973460279, −7.932345607793369842201101069293, −7.15243749921194784721603852202, −6.58856928297867522915425130966, −5.00274941501103980711444190520, −3.63930121006975049587152770585, −1.81807189040961906197830915950, 1.13428175411469359210137663140, 2.91719669066742258921296193121, 4.47473126647408572043233395744, 5.72178099945910399234259131653, 6.41455485550685883089827964218, 8.304413010388482512000570556571, 8.876625735899658693110032411474, 10.00324040944905572066418597255, 10.86602982899103345212694828585, 11.68600545758919832450359608337

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