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Properties

Label 2-273-91.45-c1-0-14
Degree $2$
Conductor $273$
Sign $0.653 + 0.757i$
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  + (0.556 − 0.556i)2-s + (0.866 − 0.5i)3-s + 1.38i·4-s + (0.542 − 2.02i)5-s + (0.203 − 0.759i)6-s + (−0.405 − 2.61i)7-s + (1.88 + 1.88i)8-s + (0.499 − 0.866i)9-s + (−0.824 − 1.42i)10-s + (−0.632 + 2.36i)11-s + (0.690 + 1.19i)12-s + (1.96 − 3.02i)13-s + (−1.67 − 1.22i)14-s + (−0.542 − 2.02i)15-s − 0.671·16-s + 6.55·17-s + ⋯
L(s)  = 1  + (0.393 − 0.393i)2-s + (0.499 − 0.288i)3-s + 0.690i·4-s + (0.242 − 0.906i)5-s + (0.0830 − 0.310i)6-s + (−0.153 − 0.988i)7-s + (0.664 + 0.664i)8-s + (0.166 − 0.288i)9-s + (−0.260 − 0.451i)10-s + (−0.190 + 0.712i)11-s + (0.199 + 0.345i)12-s + (0.546 − 0.837i)13-s + (−0.448 − 0.328i)14-s + (−0.140 − 0.523i)15-s − 0.167·16-s + 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.653 + 0.757i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.653 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69436 - 0.776116i\)
\(L(\frac12)\) \(\approx\) \(1.69436 - 0.776116i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.405 + 2.61i)T \)
13 \( 1 + (-1.96 + 3.02i)T \)
good2 \( 1 + (-0.556 + 0.556i)T - 2iT^{2} \)
5 \( 1 + (-0.542 + 2.02i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.632 - 2.36i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 6.55T + 17T^{2} \)
19 \( 1 + (7.74 - 2.07i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 3.84iT - 23T^{2} \)
29 \( 1 + (1.25 - 2.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.457 - 0.122i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (5.00 + 5.00i)T + 37iT^{2} \)
41 \( 1 + (11.0 - 2.94i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.810 + 0.467i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.03 - 1.88i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.08 - 1.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.92 - 3.92i)T - 59iT^{2} \)
61 \( 1 + (-8.13 - 4.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.1 + 2.98i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.44 - 2.52i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.62 - 9.79i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.07 - 1.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.52 - 1.52i)T + 83iT^{2} \)
89 \( 1 + (-4.45 + 4.45i)T - 89iT^{2} \)
97 \( 1 + (-4.17 + 15.5i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.24699409267173106254828985559, −10.81558506179213504308130021369, −9.995501885027585660923941176050, −8.704464121000140420719182091736, −7.950464944034637765297258887402, −7.11066140379130614931802650649, −5.45947091830795806648804464408, −4.21626743598934391790762256920, −3.29663464048886981242137907355, −1.58875421113835331563360601882, 2.17852510387696265313145166702, 3.54442447243059153329298338317, 5.00804534531329238445099513500, 6.13830741170557474962183875642, 6.72384091240921797444657965525, 8.274139316967357945864912316918, 9.159006732179488674825827323100, 10.26654511276661294617135546384, 10.81368897912702003842730871996, 12.05364991167127393675968986561

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