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Properties

Label 2-273-91.89-c1-0-15
Degree $2$
Conductor $273$
Sign $0.562 + 0.826i$
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  + (1.07 + 1.07i)2-s + (−0.866 − 0.5i)3-s + 0.313i·4-s + (−0.962 − 3.59i)5-s + (−0.393 − 1.46i)6-s + (−2.21 + 1.44i)7-s + (1.81 − 1.81i)8-s + (0.499 + 0.866i)9-s + (2.82 − 4.89i)10-s + (−0.516 − 1.92i)11-s + (0.156 − 0.271i)12-s + (1.45 − 3.29i)13-s + (−3.93 − 0.825i)14-s + (−0.962 + 3.59i)15-s + 4.52·16-s + 4.72·17-s + ⋯
L(s)  = 1  + (0.760 + 0.760i)2-s + (−0.499 − 0.288i)3-s + 0.156i·4-s + (−0.430 − 1.60i)5-s + (−0.160 − 0.599i)6-s + (−0.837 + 0.546i)7-s + (0.641 − 0.641i)8-s + (0.166 + 0.288i)9-s + (0.894 − 1.54i)10-s + (−0.155 − 0.581i)11-s + (0.0453 − 0.0784i)12-s + (0.404 − 0.914i)13-s + (−1.05 − 0.220i)14-s + (−0.248 + 0.927i)15-s + 1.13·16-s + 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.562 + 0.826i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.562 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19433 - 0.631858i\)
\(L(\frac12)\) \(\approx\) \(1.19433 - 0.631858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (2.21 - 1.44i)T \)
13 \( 1 + (-1.45 + 3.29i)T \)
good2 \( 1 + (-1.07 - 1.07i)T + 2iT^{2} \)
5 \( 1 + (0.962 + 3.59i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.516 + 1.92i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 4.72T + 17T^{2} \)
19 \( 1 + (4.30 + 1.15i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 2.08iT - 23T^{2} \)
29 \( 1 + (-5.10 - 8.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.20 + 0.591i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.455 + 0.455i)T - 37iT^{2} \)
41 \( 1 + (-8.81 - 2.36i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.0966 - 0.0557i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.33 - 1.16i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.18 + 3.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.19 + 1.19i)T + 59iT^{2} \)
61 \( 1 + (-5.84 + 3.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.87 + 0.771i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-14.4 + 3.87i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.72 - 10.1i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.0273 - 0.0473i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.05 + 9.05i)T - 83iT^{2} \)
89 \( 1 + (5.06 + 5.06i)T + 89iT^{2} \)
97 \( 1 + (-3.20 - 11.9i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.31992160290905538592658898906, −10.94788303142673476614581152928, −9.785607838670310372204465567081, −8.647462824940632224946191034676, −7.77926867331440824589364421351, −6.43263373413236959736125091079, −5.56952969962267124155938311507, −4.97299897964915971652993270173, −3.58807569523065818002972317021, −0.924466282168888674082190792485, 2.51048090155427465101195029474, 3.65483984444437875273802656899, 4.32630059638858435235246320893, 6.08172221294045711713221215188, 6.93455919960908137637096910833, 7.927037554234680105151265671734, 9.785951127924601868433304495789, 10.49245538701064526010761838400, 11.13198308424703556331561366253, 11.98963582084171390699032985178

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