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Properties

Label 2-273-91.24-c1-0-0
Degree $2$
Conductor $273$
Sign $-0.954 + 0.296i$
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  + (−2.38 + 0.638i)2-s + i·3-s + (3.53 − 2.04i)4-s + (0.488 + 0.130i)5-s + (−0.638 − 2.38i)6-s + (−2.15 − 1.53i)7-s + (−3.63 + 3.63i)8-s − 9-s − 1.24·10-s + (−4.20 + 4.20i)11-s + (2.04 + 3.53i)12-s + (3.50 − 0.833i)13-s + (6.11 + 2.29i)14-s + (−0.130 + 0.488i)15-s + (2.25 − 3.91i)16-s + (0.466 + 0.808i)17-s + ⋯
L(s)  = 1  + (−1.68 + 0.451i)2-s + 0.577i·3-s + (1.76 − 1.02i)4-s + (0.218 + 0.0585i)5-s + (−0.260 − 0.972i)6-s + (−0.813 − 0.581i)7-s + (−1.28 + 1.28i)8-s − 0.333·9-s − 0.394·10-s + (−1.26 + 1.26i)11-s + (0.589 + 1.02i)12-s + (0.972 − 0.231i)13-s + (1.63 + 0.613i)14-s + (−0.0337 + 0.126i)15-s + (0.564 − 0.978i)16-s + (0.113 + 0.196i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.954 + 0.296i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.954 + 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0240470 - 0.158402i\)
\(L(\frac12)\) \(\approx\) \(0.0240470 - 0.158402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 + 1.53i)T \)
13 \( 1 + (-3.50 + 0.833i)T \)
good2 \( 1 + (2.38 - 0.638i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.488 - 0.130i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (4.20 - 4.20i)T - 11iT^{2} \)
17 \( 1 + (-0.466 - 0.808i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.60 - 5.60i)T - 19iT^{2} \)
23 \( 1 + (7.03 + 4.06i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.96 - 3.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.636 + 2.37i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.314 - 1.17i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.84 + 2.10i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.152 + 0.0881i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.444 + 1.65i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.750 + 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.10 + 4.13i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 7.79iT - 61T^{2} \)
67 \( 1 + (-5.39 - 5.39i)T + 67iT^{2} \)
71 \( 1 + (-6.54 + 1.75i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (12.2 - 3.27i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.64 - 8.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.66 - 1.66i)T - 83iT^{2} \)
89 \( 1 + (-4.12 + 1.10i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.856 - 3.19i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.23861797259374615313236413403, −10.69529554929285258134585278240, −10.19616753362509192654570261979, −9.870842480958296574909238061138, −8.502301724759754959430298392253, −7.904999917800350658016307566446, −6.70217561227105828985495965663, −5.84876069949248486769226210130, −4.03584904607966454208668152733, −2.12377829217990540406905438498, 0.19195685194848644875743699873, 2.07872391033040046721571175457, 3.19576489897110192686831496764, 5.76957283128909515569521065502, 6.62789764660312530226573455326, 7.931123494328153280761278706681, 8.543193910864734479616568492547, 9.362943785896986169834196448402, 10.37889804489904501493044709150, 11.18097452241988350773003183921

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