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Properties

Label 2-273-13.10-c1-0-13
Degree $2$
Conductor $273$
Sign $0.607 + 0.794i$
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  + (2.35 − 1.36i)2-s + (0.5 + 0.866i)3-s + (2.70 − 4.68i)4-s + 1.58i·5-s + (2.35 + 1.36i)6-s + (−0.866 − 0.5i)7-s − 9.26i·8-s + (−0.499 + 0.866i)9-s + (2.15 + 3.72i)10-s + (−4.79 + 2.77i)11-s + 5.40·12-s + (−1.34 + 3.34i)13-s − 2.72·14-s + (−1.37 + 0.791i)15-s + (−7.20 − 12.4i)16-s + (2.40 − 4.16i)17-s + ⋯
L(s)  = 1  + (1.66 − 0.962i)2-s + (0.288 + 0.499i)3-s + (1.35 − 2.34i)4-s + 0.707i·5-s + (0.962 + 0.555i)6-s + (−0.327 − 0.188i)7-s − 3.27i·8-s + (−0.166 + 0.288i)9-s + (0.680 + 1.17i)10-s + (−1.44 + 0.835i)11-s + 1.56·12-s + (−0.372 + 0.928i)13-s − 0.727·14-s + (−0.353 + 0.204i)15-s + (−1.80 − 3.11i)16-s + (0.582 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.607 + 0.794i$
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: \((2,\ 273,\ (\ :1/2),\ 0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.71719 - 1.34257i\)
\(L(\frac12)\) \(\approx\) \(2.71719 - 1.34257i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 - 3.34i)T \)
good2 \( 1 + (-2.35 + 1.36i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 1.58iT - 5T^{2} \)
11 \( 1 + (4.79 - 2.77i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.40 + 4.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.772 - 0.446i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.31 + 4.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.941 - 1.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.47iT - 31T^{2} \)
37 \( 1 + (-4.32 + 2.49i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.45 - 4.87i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.506 + 0.877i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.5iT - 47T^{2} \)
53 \( 1 - 6.25T + 53T^{2} \)
59 \( 1 + (-2.98 - 1.72i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.79 + 3.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.7 + 7.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.3 + 6.56i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.33iT - 73T^{2} \)
79 \( 1 - 0.779T + 79T^{2} \)
83 \( 1 + 1.47iT - 83T^{2} \)
89 \( 1 + (-6.63 + 3.82i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.657 + 0.379i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.89261668566078836418775393773, −10.91599322798410492088012813778, −10.21390803795710241310597094834, −9.575978456717482703278306121680, −7.48207571821810702429236030062, −6.48672961415589771451950149702, −5.16726812602374416702273315418, −4.42398126035047326739070625252, −3.11055545909493115274141146038, −2.35216438680524340918771762880, 2.67669109315649238727974834335, 3.69861323246855202456510593678, 5.31533144455366152013608078348, 5.62833198717338760077453357934, 6.94301012493562220297007198887, 8.038688518402649053249211895784, 8.417531287306926402136886264660, 10.33742669728990178093766645580, 11.72540932755023456085980411506, 12.51350698249334912478175911190

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