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Properties

Label 2-273-13.10-c1-0-15
Degree $2$
Conductor $273$
Sign $-0.355 + 0.934i$
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  + (1.72 − 0.993i)2-s + (−0.5 − 0.866i)3-s + (0.975 − 1.68i)4-s − 2.85i·5-s + (−1.72 − 0.993i)6-s + (−0.866 − 0.5i)7-s + 0.0979i·8-s + (−0.499 + 0.866i)9-s + (−2.83 − 4.91i)10-s + (−1.99 + 1.15i)11-s − 1.95·12-s + (3.55 − 0.620i)13-s − 1.98·14-s + (−2.47 + 1.42i)15-s + (2.04 + 3.54i)16-s + (1.26 − 2.18i)17-s + ⋯
L(s)  = 1  + (1.21 − 0.702i)2-s + (−0.288 − 0.499i)3-s + (0.487 − 0.844i)4-s − 1.27i·5-s + (−0.702 − 0.405i)6-s + (−0.327 − 0.188i)7-s + 0.0346i·8-s + (−0.166 + 0.288i)9-s + (−0.896 − 1.55i)10-s + (−0.602 + 0.347i)11-s − 0.563·12-s + (0.985 − 0.172i)13-s − 0.531·14-s + (−0.637 + 0.368i)15-s + (0.512 + 0.886i)16-s + (0.306 − 0.530i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.355 + 0.934i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.355 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16261 - 1.68601i\)
\(L(\frac12)\) \(\approx\) \(1.16261 - 1.68601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-3.55 + 0.620i)T \)
good2 \( 1 + (-1.72 + 0.993i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 2.85iT - 5T^{2} \)
11 \( 1 + (1.99 - 1.15i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.26 + 2.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.98 - 2.30i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.05 + 3.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.34 - 2.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.917iT - 31T^{2} \)
37 \( 1 + (-5.14 + 2.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.59 - 2.07i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.33 - 9.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.601iT - 47T^{2} \)
53 \( 1 - 1.41T + 53T^{2} \)
59 \( 1 + (4.22 + 2.43i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.72 - 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.3 - 7.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.6 - 6.15i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 - 6.67T + 79T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + (9.36 - 5.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.46 - 2.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.02854212766152039271082625260, −11.05766292714691165389981597892, −9.983085382734161959550880318185, −8.665314079859213489127677371097, −7.73030446634053093547610724728, −6.15024194586159821583773677317, −5.25719467262691477678159254767, −4.39748784442690455742335830359, −3.02866207562938846791269847982, −1.34155023466227341585228271325, 3.05453953878134623888870215124, 3.81430125949280635106310029079, 5.25963817651951677676932116595, 6.09445558465383114535799460825, 6.82990509931457157159292784104, 7.972265028943888722523881638482, 9.536048603432889651892371867996, 10.47316215605954370066812411832, 11.32334062714117957171093734538, 12.27805918800910841694198918707

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