Properties

Label 2-273-273.269-c1-0-8
Degree $2$
Conductor $273$
Sign $0.998 - 0.0497i$
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  + 0.546i·2-s + (−1.72 − 0.117i)3-s + 1.70·4-s + (−1.25 − 2.17i)5-s + (0.0643 − 0.943i)6-s + (−0.612 + 2.57i)7-s + 2.02i·8-s + (2.97 + 0.407i)9-s + (1.18 − 0.686i)10-s + (5.22 − 3.01i)11-s + (−2.94 − 0.200i)12-s + (1.89 − 3.06i)13-s + (−1.40 − 0.334i)14-s + (1.91 + 3.90i)15-s + 2.29·16-s + 0.647·17-s + ⋯
L(s)  = 1  + 0.386i·2-s + (−0.997 − 0.0680i)3-s + 0.850·4-s + (−0.561 − 0.973i)5-s + (0.0262 − 0.385i)6-s + (−0.231 + 0.972i)7-s + 0.714i·8-s + (0.990 + 0.135i)9-s + (0.375 − 0.217i)10-s + (1.57 − 0.908i)11-s + (−0.848 − 0.0579i)12-s + (0.526 − 0.850i)13-s + (−0.375 − 0.0894i)14-s + (0.494 + 1.00i)15-s + 0.574·16-s + 0.156·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.998 - 0.0497i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.998 - 0.0497i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14162 + 0.0284411i\)
\(L(\frac12)\) \(\approx\) \(1.14162 + 0.0284411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.117i)T \)
7 \( 1 + (0.612 - 2.57i)T \)
13 \( 1 + (-1.89 + 3.06i)T \)
good2 \( 1 - 0.546iT - 2T^{2} \)
5 \( 1 + (1.25 + 2.17i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.22 + 3.01i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.647T + 17T^{2} \)
19 \( 1 + (-1.98 - 1.14i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.03iT - 23T^{2} \)
29 \( 1 + (4.46 + 2.57i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.25 - 5.34i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.95T + 37T^{2} \)
41 \( 1 + (4.48 - 7.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.09 + 7.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.09 + 3.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0406 + 0.0234i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 + (3.11 + 1.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.761 + 1.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.4 - 6.62i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.76 - 2.75i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.14 + 3.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 1.01T + 89T^{2} \)
97 \( 1 + (8.58 - 4.95i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.85392257168861862239731739332, −11.39913821683476201788688217008, −10.13862719615405359122623054351, −8.808474087247963317459974721809, −8.057836985926602331570348017554, −6.67155699802952127289634736870, −5.96039977024444383116157192128, −5.09944294027900314414649353509, −3.48926993478795471455705295416, −1.26804323383487103868696715028, 1.46748252370581631838195231170, 3.50705792284743348659747024857, 4.34705405631353394375081884466, 6.34322733865003574507879834909, 6.83385493809338824141358350770, 7.45420939744281520268750655293, 9.511166988885278413642598106583, 10.30159754679929098747312606829, 11.16832353566685011979798372905, 11.63624679304217676424204153358

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