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Properties

Label 2-273-273.68-c1-0-24
Degree $2$
Conductor $273$
Sign $-0.331 + 0.943i$
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  − 1.97i·2-s + (1.69 − 0.372i)3-s − 1.88·4-s + (−0.272 + 0.472i)5-s + (−0.735 − 3.33i)6-s + (2.35 + 1.20i)7-s − 0.226i·8-s + (2.72 − 1.26i)9-s + (0.930 + 0.537i)10-s + (0.383 + 0.221i)11-s + (−3.18 + 0.703i)12-s + (−2.64 − 2.45i)13-s + (2.37 − 4.64i)14-s + (−0.284 + 0.900i)15-s − 4.21·16-s − 3.83·17-s + ⋯
L(s)  = 1  − 1.39i·2-s + (0.976 − 0.215i)3-s − 0.942·4-s + (−0.121 + 0.211i)5-s + (−0.300 − 1.36i)6-s + (0.890 + 0.454i)7-s − 0.0801i·8-s + (0.907 − 0.420i)9-s + (0.294 + 0.169i)10-s + (0.115 + 0.0667i)11-s + (−0.920 + 0.202i)12-s + (−0.732 − 0.680i)13-s + (0.633 − 1.24i)14-s + (−0.0735 + 0.232i)15-s − 1.05·16-s − 0.931·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.331 + 0.943i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02289 - 1.44436i\)
\(L(\frac12)\) \(\approx\) \(1.02289 - 1.44436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 + 0.372i)T \)
7 \( 1 + (-2.35 - 1.20i)T \)
13 \( 1 + (2.64 + 2.45i)T \)
good2 \( 1 + 1.97iT - 2T^{2} \)
5 \( 1 + (0.272 - 0.472i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.383 - 0.221i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.83T + 17T^{2} \)
19 \( 1 + (-0.0717 + 0.0414i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.16iT - 23T^{2} \)
29 \( 1 + (3.66 - 2.11i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.32 + 3.07i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.94T + 37T^{2} \)
41 \( 1 + (2.97 + 5.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.19 - 5.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.322 - 0.559i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.39 + 4.26i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.84T + 59T^{2} \)
61 \( 1 + (4.76 - 2.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.03 + 8.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.56 - 4.36i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-12.7 + 7.37i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.78 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.87T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (-1.99 - 1.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.60294817844651789654568128505, −10.81242977283159790894674453994, −9.747751821892581524081984761596, −9.024498491387047682517587877877, −7.945043302825200158771322406749, −6.95246155980604341897237764836, −5.05540440084715597355499369703, −3.76939661388666352193291748370, −2.68151797694165016022893903843, −1.63864838529599196388463169436, 2.26823862176235028780821081799, 4.32406855370373520250116359368, 4.88833456186621604824979192116, 6.60012555808286860990071525846, 7.30653842408062592067487945354, 8.384533328389227341410721469430, 8.720283204722810982152161534449, 10.03054332190077528004283745822, 11.14902075885807196980071673295, 12.38351138734172420559557609053

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