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Properties

Label 2-273-91.25-c1-0-17
Degree $2$
Conductor $273$
Sign $-0.0896 + 0.995i$
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  + (2.35 − 1.36i)2-s + (0.5 − 0.866i)3-s + (2.71 − 4.69i)4-s + (−1.84 + 1.06i)5-s − 2.72i·6-s + (−1.72 + 2.00i)7-s − 9.32i·8-s + (−0.499 − 0.866i)9-s + (−2.89 + 5.01i)10-s + (3.75 + 2.16i)11-s + (−2.71 − 4.69i)12-s + (−1.28 + 3.36i)13-s + (−1.33 + 7.08i)14-s + 2.12i·15-s + (−7.28 − 12.6i)16-s + (0.109 − 0.190i)17-s + ⋯
L(s)  = 1  + (1.66 − 0.963i)2-s + (0.288 − 0.499i)3-s + (1.35 − 2.34i)4-s + (−0.823 + 0.475i)5-s − 1.11i·6-s + (−0.651 + 0.758i)7-s − 3.29i·8-s + (−0.166 − 0.288i)9-s + (−0.916 + 1.58i)10-s + (1.13 + 0.653i)11-s + (−0.782 − 1.35i)12-s + (−0.355 + 0.934i)13-s + (−0.356 + 1.89i)14-s + 0.549i·15-s + (−1.82 − 3.15i)16-s + (0.0266 − 0.0460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0896 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0896 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0896 + 0.995i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0896 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93957 - 2.12197i\)
\(L(\frac12)\) \(\approx\) \(1.93957 - 2.12197i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 2.00i)T \)
13 \( 1 + (1.28 - 3.36i)T \)
good2 \( 1 + (-2.35 + 1.36i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.84 - 1.06i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.75 - 2.16i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.109 + 0.190i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.42 + 2.55i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0830 + 0.143i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.18T + 29T^{2} \)
31 \( 1 + (5.78 + 3.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.92 - 5.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.73iT - 41T^{2} \)
43 \( 1 + 3.96T + 43T^{2} \)
47 \( 1 + (2.30 - 1.33i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.75 + 9.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.36 - 0.785i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.23 - 7.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.00 + 2.89i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.29iT - 71T^{2} \)
73 \( 1 + (6.44 + 3.71i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.46 - 14.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.46iT - 83T^{2} \)
89 \( 1 + (-10.7 + 6.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.414iT - 97T^{2} \)
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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

−11.75258866765553232316286755333, −11.47833415352021915742162583049, −9.977175782267894043488068322065, −9.123967505193594405372392432187, −7.10760893225649938067832980616, −6.62451155447568458239794366646, −5.29560903219240016180620204636, −3.99695131546541041398887659990, −3.15752184381676238767008287889, −1.89954046425184566574788998369, 3.37657121088567950970190153645, 3.75005482467819222789976805749, 4.90532324689475815750509389427, 5.95834723806194518408896622400, 7.13111823025654134132065606848, 7.88783842031743402455706108836, 8.938674474619368536006505282027, 10.50680625922120958246410185062, 11.72236892185922175806526935669, 12.32897693313609330977565408717

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