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Properties

Label 2-273-273.251-c1-0-13
Degree $2$
Conductor $273$
Sign $0.916 + 0.400i$
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  + (0.376 − 0.652i)2-s + (−0.798 − 1.53i)3-s + (0.716 + 1.24i)4-s + 2.14i·5-s + (−1.30 − 0.0577i)6-s + (2.55 − 0.685i)7-s + 2.58·8-s + (−1.72 + 2.45i)9-s + (1.40 + 0.808i)10-s + (0.0743 − 0.128i)11-s + (1.33 − 2.09i)12-s + (3.58 + 0.428i)13-s + (0.515 − 1.92i)14-s + (3.30 − 1.71i)15-s + (−0.458 + 0.793i)16-s + (−2.50 − 4.34i)17-s + ⋯
L(s)  = 1  + (0.266 − 0.461i)2-s + (−0.461 − 0.887i)3-s + (0.358 + 0.620i)4-s + 0.960i·5-s + (−0.532 − 0.0235i)6-s + (0.965 − 0.259i)7-s + 0.914·8-s + (−0.574 + 0.818i)9-s + (0.443 + 0.255i)10-s + (0.0224 − 0.0388i)11-s + (0.385 − 0.603i)12-s + (0.992 + 0.118i)13-s + (0.137 − 0.514i)14-s + (0.852 − 0.442i)15-s + (−0.114 + 0.198i)16-s + (−0.608 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50176 - 0.313853i\)
\(L(\frac12)\) \(\approx\) \(1.50176 - 0.313853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.798 + 1.53i)T \)
7 \( 1 + (-2.55 + 0.685i)T \)
13 \( 1 + (-3.58 - 0.428i)T \)
good2 \( 1 + (-0.376 + 0.652i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.14iT - 5T^{2} \)
11 \( 1 + (-0.0743 + 0.128i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.50 + 4.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.254 - 0.441i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.305 - 0.176i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.97 + 4.60i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.08T + 31T^{2} \)
37 \( 1 + (7.26 + 4.19i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.21 + 1.28i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.758 - 1.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.0iT - 47T^{2} \)
53 \( 1 - 8.83iT - 53T^{2} \)
59 \( 1 + (-4.86 + 2.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.33 - 4.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.50 + 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.52 + 4.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.34T + 73T^{2} \)
79 \( 1 + 8.43T + 79T^{2} \)
83 \( 1 + 13.0iT - 83T^{2} \)
89 \( 1 + (10.1 + 5.83i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.54 + 13.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.59159291166748869655668909595, −11.20343093039445575737664167419, −10.57664428355452163292643114008, −8.739583503309382953990317531485, −7.59338722804657496804013789588, −7.12341147475792206211230236890, −5.95945197591308146593355676660, −4.45693983487113231522785056739, −2.99509913048394442117820674025, −1.76549692273117199176605044714, 1.51865971799396060280751449437, 3.98154407946950890855177972166, 5.04450981427046914289427599382, 5.59375254536229892981470733191, 6.73520302199968659187880011919, 8.334767039975043297569725861059, 8.968667145792193488343783736017, 10.27516672072699691664611380829, 10.98820501905728599807529029694, 11.72761073927911791551414951272

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