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Properties

Label 2-273-273.152-c1-0-13
Degree $2$
Conductor $273$
Sign $0.256 - 0.966i$
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  + (0.473 − 0.273i)2-s + (−0.966 + 1.43i)3-s + (−0.850 + 1.47i)4-s + (1.25 − 2.17i)5-s + (−0.0643 + 0.943i)6-s + (2.53 + 0.756i)7-s + 2.02i·8-s + (−1.13 − 2.77i)9-s − 1.37i·10-s + 6.02i·11-s + (−1.29 − 2.64i)12-s + (−1.89 + 3.06i)13-s + (1.40 − 0.334i)14-s + (1.91 + 3.90i)15-s + (−1.14 − 1.99i)16-s + (0.323 − 0.560i)17-s + ⋯
L(s)  = 1  + (0.334 − 0.193i)2-s + (−0.557 + 0.829i)3-s + (−0.425 + 0.736i)4-s + (0.561 − 0.973i)5-s + (−0.0262 + 0.385i)6-s + (0.958 + 0.285i)7-s + 0.714i·8-s + (−0.377 − 0.925i)9-s − 0.434i·10-s + 1.81i·11-s + (−0.374 − 0.764i)12-s + (−0.526 + 0.850i)13-s + (0.375 − 0.0894i)14-s + (0.494 + 1.00i)15-s + (−0.287 − 0.497i)16-s + (0.0784 − 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.256 - 0.966i$
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: \((2,\ 273,\ (\ :1/2),\ 0.256 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01223 + 0.778261i\)
\(L(\frac12)\) \(\approx\) \(1.01223 + 0.778261i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.966 - 1.43i)T \)
7 \( 1 + (-2.53 - 0.756i)T \)
13 \( 1 + (1.89 - 3.06i)T \)
good2 \( 1 + (-0.473 + 0.273i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.25 + 2.17i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 6.02iT - 11T^{2} \)
17 \( 1 + (-0.323 + 0.560i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 2.29iT - 19T^{2} \)
23 \( 1 + (-0.894 + 0.516i)T + (11.5 - 19.9i)T^{2} \)
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 \( 1 + (-1.97 - 3.42i)T + (-18.5 + 32.0i)T^{2} \)
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 \( 1 + (-0.831 + 1.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 3.59iT - 61T^{2} \)
67 \( 1 - 1.52T + 67T^{2} \)
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 \( 1 + (0.507 + 0.879i)T + (-44.5 + 77.0i)T^{2} \)
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

−12.07039775423312006208690549059, −11.51854046947936892376712889650, −10.02331210662978971166740303033, −9.360704777655031095749613190926, −8.492068005041048560855967184824, −7.26342773539017948599193256037, −5.55355496021277882917720204204, −4.68970532582927706400915845140, −4.24744015110425233995793733626, −2.11736243914651283813108832026, 1.05357500216640123469573638092, 2.88762482840346823455538249770, 4.85315497623782360788171106318, 5.79600892682723426112891410002, 6.44405843567065893267056279724, 7.61630853645555660404879804138, 8.663055912344519497505355956919, 10.18372015671083965860093360565, 10.85812772541271107189297776299, 11.46699584452518525293050134995

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