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Properties

Label 2-273-91.30-c1-0-4
Degree $2$
Conductor $273$
Sign $0.0103 - 0.999i$
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  + (−0.792 + 0.457i)2-s − 3-s + (−0.581 + 1.00i)4-s + (2.26 + 1.30i)5-s + (0.792 − 0.457i)6-s + (1.81 − 1.92i)7-s − 2.89i·8-s + 9-s − 2.39·10-s + 2.54i·11-s + (0.581 − 1.00i)12-s + (0.523 + 3.56i)13-s + (−0.557 + 2.35i)14-s + (−2.26 − 1.30i)15-s + (0.162 + 0.281i)16-s + (−3.09 + 5.36i)17-s + ⋯
L(s)  = 1  + (−0.560 + 0.323i)2-s − 0.577·3-s + (−0.290 + 0.503i)4-s + (1.01 + 0.585i)5-s + (0.323 − 0.186i)6-s + (0.685 − 0.727i)7-s − 1.02i·8-s + 0.333·9-s − 0.757·10-s + 0.767i·11-s + (0.167 − 0.290i)12-s + (0.145 + 0.989i)13-s + (−0.148 + 0.629i)14-s + (−0.585 − 0.337i)15-s + (0.0406 + 0.0704i)16-s + (−0.751 + 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0103 - 0.999i$
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: \((2,\ 273,\ (\ :1/2),\ 0.0103 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.624661 + 0.618229i\)
\(L(\frac12)\) \(\approx\) \(0.624661 + 0.618229i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 1.92i)T \)
13 \( 1 + (-0.523 - 3.56i)T \)
good2 \( 1 + (0.792 - 0.457i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.26 - 1.30i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.54iT - 11T^{2} \)
17 \( 1 + (3.09 - 5.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
23 \( 1 + (-3.94 - 6.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.77 - 3.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.91 + 4.56i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.52 - 4.34i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.98 + 2.30i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.11 + 3.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.53 - 0.887i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.42 + 5.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.95 - 1.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 - 0.619iT - 67T^{2} \)
71 \( 1 + (-6.48 + 3.74i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.8 + 6.24i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.621 + 1.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.10iT - 83T^{2} \)
89 \( 1 + (-9.87 + 5.70i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.32 - 0.767i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.01999338407328728009864944590, −11.02734457885220767731138337988, −10.15393808689844298482493824908, −9.377508922817039662866661643536, −8.271305775423821892550386653477, −7.02457417011793307861479184262, −6.59696403052916987261892035025, −5.00470052094963458795073790640, −3.87620276362015381790058070005, −1.77619813316759994983146140643, 0.960668296073142627047688724722, 2.43449012766365981656647271529, 4.94516945636725419616097033504, 5.39012379395932481524546374515, 6.40191614159918108685020917944, 8.251915967170102433040922277482, 8.883693763390873708481816944040, 9.808053006697337768540351060127, 10.67397434312721456488622772961, 11.43253347250931653728811995408

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