Properties

Label 2-273-273.152-c1-0-0
Degree $2$
Conductor $273$
Sign $0.188 + 0.982i$
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  + (−1.66 + 0.961i)2-s + (−0.628 + 1.61i)3-s + (0.850 − 1.47i)4-s + (−1.94 + 3.37i)5-s + (−0.506 − 3.29i)6-s + (−1.91 − 1.83i)7-s − 0.575i·8-s + (−2.21 − 2.02i)9-s − 7.48i·10-s + 2.46i·11-s + (1.84 + 2.29i)12-s + (0.534 + 3.56i)13-s + (4.94 + 1.21i)14-s + (−4.21 − 5.25i)15-s + (2.25 + 3.90i)16-s + (3.38 − 5.87i)17-s + ⋯
L(s)  = 1  + (−1.17 + 0.680i)2-s + (−0.362 + 0.931i)3-s + (0.425 − 0.736i)4-s + (−0.870 + 1.50i)5-s + (−0.206 − 1.34i)6-s + (−0.721 − 0.691i)7-s − 0.203i·8-s + (−0.737 − 0.675i)9-s − 2.36i·10-s + 0.742i·11-s + (0.532 + 0.663i)12-s + (0.148 + 0.988i)13-s + (1.32 + 0.324i)14-s + (−1.08 − 1.35i)15-s + (0.563 + 0.976i)16-s + (0.822 − 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.188 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0978901 - 0.0808977i\)
\(L(\frac12)\) \(\approx\) \(0.0978901 - 0.0808977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.628 - 1.61i)T \)
7 \( 1 + (1.91 + 1.83i)T \)
13 \( 1 + (-0.534 - 3.56i)T \)
good2 \( 1 + (1.66 - 0.961i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.94 - 3.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.46iT - 11T^{2} \)
17 \( 1 + (-3.38 + 5.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 2.68iT - 19T^{2} \)
23 \( 1 + (2.41 - 1.39i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0349 - 0.0201i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.270 - 0.156i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.03 + 3.53i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.73 + 3.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.35 + 9.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.67 - 4.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.37 - 2.52i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.73 - 3.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.79iT - 61T^{2} \)
67 \( 1 + 6.47T + 67T^{2} \)
71 \( 1 + (8.84 - 5.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.25 - 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.62 + 4.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.83T + 83T^{2} \)
89 \( 1 + (5.20 + 9.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.5 + 7.27i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.18595372136726417507789628392, −11.36754879232725583159300192442, −10.28661689019429380206164940543, −9.971421378561294192785608780215, −8.984957960212504674454537967955, −7.49358500993572539173240654141, −7.10889856926507487032207153828, −6.11736700917915418041350705536, −4.18415152283349980889430527423, −3.29159350487469743880546812851, 0.15916169117324792593093016926, 1.38914379962186639979591828406, 3.17124567949000980237140991171, 5.16587824688886095218556259593, 6.12240961151964427014850534247, 7.944210226786491262790944217279, 8.253038505609882546366133944109, 9.027363927277353801279576233939, 10.21359389156284191498143653627, 11.30395309874734033784811738422

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