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Properties

Label 2-273-7.4-c1-0-7
Degree $2$
Conductor $273$
Sign $0.678 - 0.734i$
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  + (0.766 + 1.32i)2-s + (0.5 − 0.866i)3-s + (−0.173 + 0.300i)4-s + (0.266 + 0.460i)5-s + 1.53·6-s + (−0.418 + 2.61i)7-s + 2.53·8-s + (−0.499 − 0.866i)9-s + (−0.407 + 0.705i)10-s + (1.43 − 2.49i)11-s + (0.173 + 0.300i)12-s + 13-s + (−3.78 + 1.44i)14-s + 0.532·15-s + (2.28 + 3.96i)16-s + (−1.67 + 2.89i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.938i)2-s + (0.288 − 0.499i)3-s + (−0.0868 + 0.150i)4-s + (0.118 + 0.206i)5-s + 0.625·6-s + (−0.158 + 0.987i)7-s + 0.895·8-s + (−0.166 − 0.288i)9-s + (−0.128 + 0.223i)10-s + (0.434 − 0.751i)11-s + (0.0501 + 0.0868i)12-s + 0.277·13-s + (−1.01 + 0.386i)14-s + 0.137·15-s + (0.571 + 0.990i)16-s + (−0.405 + 0.703i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79432 + 0.785083i\)
\(L(\frac12)\) \(\approx\) \(1.79432 + 0.785083i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.418 - 2.61i)T \)
13 \( 1 - T \)
good2 \( 1 + (-0.766 - 1.32i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.266 - 0.460i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.43 + 2.49i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.03 - 1.78i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.93 + 6.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.04T + 29T^{2} \)
31 \( 1 + (3.11 - 5.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.326 - 0.565i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.59T + 41T^{2} \)
43 \( 1 + 6.10T + 43T^{2} \)
47 \( 1 + (4.75 + 8.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.439 - 0.761i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.12 + 1.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.14 - 7.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.19 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + (-0.275 + 0.477i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.80 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.87T + 83T^{2} \)
89 \( 1 + (2.74 + 4.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−12.29610105267288106538245140514, −11.18621422496983710081030934783, −10.13270786056993683692326154864, −8.722841275944243403683594624864, −8.147205912439721051992829394950, −6.70764942363058714527513982113, −6.23162276516275759833462516153, −5.23409632077389567229641349997, −3.66877417844796264299611007698, −1.99493139200842315950946286956, 1.76776999518867029654714118961, 3.37395727014626571350323287754, 4.15094583458727162217313061058, 5.20479787124233952593703390590, 6.99941053800716363588465047657, 7.80458163713111528759279114076, 9.391153383382908894784328145263, 9.911391176372354987192699777731, 11.13214533691534374554736122158, 11.52537975651355572448986238695

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