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Properties

Label 2-273-21.5-c1-0-15
Degree $2$
Conductor $273$
Sign $0.519 + 0.854i$
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  + (−1.00 − 0.582i)2-s + (1.67 − 0.436i)3-s + (−0.320 − 0.554i)4-s + (−0.156 + 0.271i)5-s + (−1.94 − 0.536i)6-s + (2.13 + 1.55i)7-s + 3.07i·8-s + (2.61 − 1.46i)9-s + (0.316 − 0.182i)10-s + (3.79 − 2.19i)11-s + (−0.779 − 0.790i)12-s − i·13-s + (−1.25 − 2.81i)14-s + (−0.144 + 0.523i)15-s + (1.15 − 1.99i)16-s + (−1.80 − 3.11i)17-s + ⋯
L(s)  = 1  + (−0.713 − 0.412i)2-s + (0.967 − 0.252i)3-s + (−0.160 − 0.277i)4-s + (−0.0700 + 0.121i)5-s + (−0.794 − 0.218i)6-s + (0.808 + 0.588i)7-s + 1.08i·8-s + (0.872 − 0.487i)9-s + (0.100 − 0.0577i)10-s + (1.14 − 0.660i)11-s + (−0.224 − 0.228i)12-s − 0.277i·13-s + (−0.334 − 0.753i)14-s + (−0.0372 + 0.135i)15-s + (0.288 − 0.499i)16-s + (−0.436 − 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.519 + 0.854i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.519 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08483 - 0.610317i\)
\(L(\frac12)\) \(\approx\) \(1.08483 - 0.610317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 + 0.436i)T \)
7 \( 1 + (-2.13 - 1.55i)T \)
13 \( 1 + iT \)
good2 \( 1 + (1.00 + 0.582i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.156 - 0.271i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.79 + 2.19i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.80 + 3.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.27 + 0.736i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.20 + 3.58i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.70iT - 29T^{2} \)
31 \( 1 + (4.67 - 2.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.07 - 1.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.78T + 41T^{2} \)
43 \( 1 + 0.0932T + 43T^{2} \)
47 \( 1 + (-3.56 + 6.17i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.29 - 5.36i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.87 - 10.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.91 + 3.41i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.49 - 9.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.39iT - 71T^{2} \)
73 \( 1 + (4.82 - 2.78i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.21 - 2.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + (3.01 - 5.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.53iT - 97T^{2} \)
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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

−11.55142611688397494202741728868, −10.77183200592799381808753924000, −9.584259582221496609314817263145, −8.832273105023463262813015439419, −8.355200847148916901246468747658, −7.10575255386881677885878806745, −5.70929290230657844513910308563, −4.28608195356775660864880658773, −2.67044343510065131295680457158, −1.40394861475904905723832403654, 1.77096420089172899525790990745, 3.89334755709987667738726057627, 4.36226937707956130722419093739, 6.50682166621052009312762748780, 7.59197626336733423116814491627, 8.150842136698962984562368060097, 9.119886460602198826530508890155, 9.766968719844990546380715694649, 10.84643387151933416598008569878, 12.15624638779023106871184360469

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