Properties

Label 2-273-273.254-c1-0-32
Degree $2$
Conductor $273$
Sign $-0.514 - 0.857i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.542 − 0.145i)2-s + (−1.41 − 1.00i)3-s + (−1.45 − 0.842i)4-s + (−0.881 − 3.29i)5-s + (0.621 + 0.748i)6-s + (−2.35 − 1.21i)7-s + (1.46 + 1.46i)8-s + (0.995 + 2.83i)9-s + 1.91i·10-s + (3.20 + 3.20i)11-s + (1.21 + 2.65i)12-s + (−1.88 + 3.07i)13-s + (1.09 + 1.00i)14-s + (−2.04 + 5.53i)15-s + (1.10 + 1.91i)16-s + (1.48 − 2.57i)17-s + ⋯
L(s)  = 1  + (−0.383 − 0.102i)2-s + (−0.816 − 0.578i)3-s + (−0.729 − 0.421i)4-s + (−0.394 − 1.47i)5-s + (0.253 + 0.305i)6-s + (−0.888 − 0.458i)7-s + (0.517 + 0.517i)8-s + (0.331 + 0.943i)9-s + 0.605i·10-s + (0.966 + 0.966i)11-s + (0.351 + 0.765i)12-s + (−0.523 + 0.852i)13-s + (0.293 + 0.267i)14-s + (−0.529 + 1.42i)15-s + (0.275 + 0.477i)16-s + (0.360 − 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.514 - 0.857i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.514 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0675700 + 0.119268i\)
\(L(\frac12)\) \(\approx\) \(0.0675700 + 0.119268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.41 + 1.00i)T \)
7 \( 1 + (2.35 + 1.21i)T \)
13 \( 1 + (1.88 - 3.07i)T \)
good2 \( 1 + (0.542 + 0.145i)T + (1.73 + i)T^{2} \)
5 \( 1 + (0.881 + 3.29i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.20 - 3.20i)T + 11iT^{2} \)
17 \( 1 + (-1.48 + 2.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.71 + 4.71i)T + 19iT^{2} \)
23 \( 1 + (1.86 + 3.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0674 - 0.0389i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.64 - 6.13i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.44 - 0.387i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.861 - 3.21i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.85 - 2.80i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.7 - 2.87i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.45 + 1.41i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.52 + 0.408i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 5.86T + 61T^{2} \)
67 \( 1 + (4.78 + 4.78i)T + 67iT^{2} \)
71 \( 1 + (12.1 + 3.25i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.99 + 0.803i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.07 - 8.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.49 + 5.49i)T - 83iT^{2} \)
89 \( 1 + (-1.93 + 7.20i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.54 + 5.78i)T + (-84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.43002966924634308363976729770, −10.13469101753393405436977050243, −9.375078611065790429306055416286, −8.588024520700664444959484019444, −7.28165231588160947195143659624, −6.33262334143623001077318407051, −4.74772273936439487748933195532, −4.49169803054010827160364316221, −1.51674419702254677636872645295, −0.14032290442754624316541683534, 3.34640155568846781985699050724, 3.90335516956513303954771454577, 5.76510904761708473824970405348, 6.48012846382437505311422561628, 7.66721482986710128170423339460, 8.829037750927623273054971905986, 9.955318655531555451961069372942, 10.39056739569248592996158884169, 11.52567478100451126166598838055, 12.30858107418165038674538037877

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