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Properties

Label 2-273-91.80-c1-0-5
Degree $2$
Conductor $273$
Sign $0.504 + 0.863i$
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  + (−0.720 − 2.68i)2-s + i·3-s + (−4.97 + 2.87i)4-s + (−0.470 + 1.75i)5-s + (2.68 − 0.720i)6-s + (1.29 − 2.30i)7-s + (7.38 + 7.38i)8-s − 9-s + 5.05·10-s + (3.13 + 3.13i)11-s + (−2.87 − 4.97i)12-s + (2.12 − 2.91i)13-s + (−7.13 − 1.81i)14-s + (−1.75 − 0.470i)15-s + (8.78 − 15.2i)16-s + (0.0321 + 0.0557i)17-s + ⋯
L(s)  = 1  + (−0.509 − 1.90i)2-s + 0.577i·3-s + (−2.48 + 1.43i)4-s + (−0.210 + 0.784i)5-s + (1.09 − 0.294i)6-s + (0.489 − 0.872i)7-s + (2.61 + 2.61i)8-s − 0.333·9-s + 1.59·10-s + (0.945 + 0.945i)11-s + (−0.829 − 1.43i)12-s + (0.590 − 0.807i)13-s + (−1.90 − 0.486i)14-s + (−0.452 − 0.121i)15-s + (2.19 − 3.80i)16-s + (0.00780 + 0.0135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.504 + 0.863i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.504 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.779391 - 0.447316i\)
\(L(\frac12)\) \(\approx\) \(0.779391 - 0.447316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-1.29 + 2.30i)T \)
13 \( 1 + (-2.12 + 2.91i)T \)
good2 \( 1 + (0.720 + 2.68i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.470 - 1.75i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.13 - 3.13i)T + 11iT^{2} \)
17 \( 1 + (-0.0321 - 0.0557i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.15 - 2.15i)T + 19iT^{2} \)
23 \( 1 + (-2.78 - 1.60i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.41 - 4.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.02 - 1.34i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-4.21 + 1.12i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.12 - 7.93i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.22 + 1.86i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.06 - 1.08i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.30 + 5.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.49 + 1.20i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 2.19iT - 61T^{2} \)
67 \( 1 + (-4.45 + 4.45i)T - 67iT^{2} \)
71 \( 1 + (1.98 + 7.40i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.61 - 6.03i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.639 - 1.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.90 + 8.90i)T + 83iT^{2} \)
89 \( 1 + (-0.376 - 1.40i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.79 - 2.35i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.35153969434730669383727894394, −10.85164224007341529603917597247, −10.12723401445139496368399301989, −9.356634987345574310571014412470, −8.256398206457793576418350130915, −7.21271210527102159918258757654, −4.97325640521922652019577073490, −3.89498006895850334325139496491, −3.15235001219902838085569303750, −1.40132255821710512781968716882, 1.06839343885753982675970448751, 4.26040568400638939405159927442, 5.40076829021570012251087545250, 6.20340705299000332385522245629, 7.12636240503533667048295957302, 8.323119111350253660259080000995, 8.766505176097071719872919306322, 9.376047317628816483756316711803, 11.12823016682140512621405967912, 12.24937471907196571303160732737

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