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Properties

Label 2-273-273.38-c1-0-1
Degree $2$
Conductor $273$
Sign $0.745 - 0.666i$
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  + (−0.903 − 1.56i)2-s + (−1.69 − 0.370i)3-s + (−0.630 + 1.09i)4-s + (−1.48 + 0.855i)5-s + (0.947 + 2.98i)6-s + (−1.44 − 2.21i)7-s − 1.33·8-s + (2.72 + 1.25i)9-s + (2.67 + 1.54i)10-s + (−1.30 + 2.25i)11-s + (1.47 − 1.61i)12-s + (3.46 + 1.01i)13-s + (−2.15 + 4.26i)14-s + (2.82 − 0.897i)15-s + (2.46 + 4.27i)16-s + (−2.88 + 4.99i)17-s + ⋯
L(s)  = 1  + (−0.638 − 1.10i)2-s + (−0.976 − 0.214i)3-s + (−0.315 + 0.546i)4-s + (−0.662 + 0.382i)5-s + (0.386 + 1.21i)6-s + (−0.547 − 0.836i)7-s − 0.471·8-s + (0.908 + 0.418i)9-s + (0.846 + 0.488i)10-s + (−0.392 + 0.680i)11-s + (0.425 − 0.466i)12-s + (0.959 + 0.280i)13-s + (−0.575 + 1.13i)14-s + (0.729 − 0.231i)15-s + (0.616 + 1.06i)16-s + (−0.698 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.745 - 0.666i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.206100 + 0.0787654i\)
\(L(\frac12)\) \(\approx\) \(0.206100 + 0.0787654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 0.370i)T \)
7 \( 1 + (1.44 + 2.21i)T \)
13 \( 1 + (-3.46 - 1.01i)T \)
good2 \( 1 + (0.903 + 1.56i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.48 - 0.855i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.30 - 2.25i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.88 - 4.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.572 + 0.991i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.58 + 3.79i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.60iT - 29T^{2} \)
31 \( 1 + (4.44 - 7.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.46 + 1.42i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + 9.39T + 43T^{2} \)
47 \( 1 + (3.97 - 2.29i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.30 + 4.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.44 + 0.834i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.6 - 6.15i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.08 - 4.09i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.702T + 71T^{2} \)
73 \( 1 + (2.18 - 3.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.412 + 0.713i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.63iT - 83T^{2} \)
89 \( 1 + (-3.87 + 2.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.61T + 97T^{2} \)
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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

−11.64097664363933103640249779545, −10.90128738569097097237447394778, −10.57993788875647012762680261607, −9.548705782702508287978455154646, −8.282103037255839959084710653286, −6.99755522634960547103602734383, −6.27422264697689673887152152948, −4.52642723677232431645841293509, −3.33932761766431661804922823665, −1.48251597969127036852994798235, 0.24326940671912659891829413344, 3.37240415231550044415125152347, 5.08645882385906171157799066435, 5.91406663968212729960186070931, 6.79097320872480557118466837713, 7.85034500465960394395913059393, 8.871690282123894787102037752189, 9.518017213368306423123998750361, 10.98457395675546062419841869455, 11.66750134245717061072180779798

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