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Properties

Label 2-273-13.9-c1-0-9
Degree $2$
Conductor $273$
Sign $-0.219 + 0.975i$
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  + (1.12 − 1.94i)2-s + (−0.5 + 0.866i)3-s + (−1.52 − 2.64i)4-s + 1.69·5-s + (1.12 + 1.94i)6-s + (−0.5 − 0.866i)7-s − 2.35·8-s + (−0.499 − 0.866i)9-s + (1.90 − 3.29i)10-s + (2.64 − 4.58i)11-s + 3.04·12-s + (−3.20 − 1.65i)13-s − 2.24·14-s + (−0.846 + 1.46i)15-s + (0.400 − 0.694i)16-s + (1.12 + 1.94i)17-s + ⋯
L(s)  = 1  + (0.794 − 1.37i)2-s + (−0.288 + 0.499i)3-s + (−0.762 − 1.32i)4-s + 0.756·5-s + (0.458 + 0.794i)6-s + (−0.188 − 0.327i)7-s − 0.833·8-s + (−0.166 − 0.288i)9-s + (0.601 − 1.04i)10-s + (0.798 − 1.38i)11-s + 0.880·12-s + (−0.888 − 0.459i)13-s − 0.600·14-s + (−0.218 + 0.378i)15-s + (0.100 − 0.173i)16-s + (0.272 + 0.471i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.219 + 0.975i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.219 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16315 - 1.45455i\)
\(L(\frac12)\) \(\approx\) \(1.16315 - 1.45455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (3.20 + 1.65i)T \)
good2 \( 1 + (-1.12 + 1.94i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.69T + 5T^{2} \)
11 \( 1 + (-2.64 + 4.58i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.12 - 1.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.74 - 6.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.38 - 5.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.78 - 6.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.89T + 31T^{2} \)
37 \( 1 + (-4.78 + 8.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.49 - 6.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.13 - 5.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.70T + 47T^{2} \)
53 \( 1 + 4.20T + 53T^{2} \)
59 \( 1 + (-3.13 - 5.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.08 + 5.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.48 + 2.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.18 - 7.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.37T + 73T^{2} \)
79 \( 1 - 5.40T + 79T^{2} \)
83 \( 1 - 0.131T + 83T^{2} \)
89 \( 1 + (-2.94 + 5.10i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.187 - 0.324i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.62840862309234314704767079639, −10.84523306048168769054630340204, −9.909797639954692552287834720194, −9.486623222551250994442705109966, −7.82745980378173426751184461032, −5.99249532090258613183453031405, −5.41166965266419713389722103024, −3.93634070671062627367646427883, −3.17708989222521120830816335471, −1.44225533239767347122338875952, 2.27925683277835292027977074577, 4.40678669092406249497700661572, 5.21797299987831338193951171058, 6.35366998535149184553058637075, 6.92539569152901660810645238942, 7.80622224410238531841527543080, 9.234483116157918548644972037406, 9.985670515563252912092888532195, 11.74970155519469893151053618960, 12.34896539940312364750324934504

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