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Properties

Label 2-273-91.33-c1-0-5
Degree $2$
Conductor $273$
Sign $-0.621 - 0.783i$
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  + (−0.273 + 1.01i)2-s + i·3-s + (0.768 + 0.443i)4-s + (1.01 + 3.80i)5-s + (−1.01 − 0.273i)6-s + (1.44 − 2.21i)7-s + (−2.15 + 2.15i)8-s − 9-s − 4.15·10-s + (0.669 − 0.669i)11-s + (−0.443 + 0.768i)12-s + (0.783 − 3.51i)13-s + (1.85 + 2.08i)14-s + (−3.80 + 1.01i)15-s + (−0.719 − 1.24i)16-s + (3.98 − 6.89i)17-s + ⋯
L(s)  = 1  + (−0.193 + 0.720i)2-s + 0.577i·3-s + (0.384 + 0.221i)4-s + (0.455 + 1.70i)5-s + (−0.415 − 0.111i)6-s + (0.547 − 0.836i)7-s + (−0.761 + 0.761i)8-s − 0.333·9-s − 1.31·10-s + (0.201 − 0.201i)11-s + (−0.128 + 0.221i)12-s + (0.217 − 0.976i)13-s + (0.496 + 0.556i)14-s + (−0.982 + 0.263i)15-s + (−0.179 − 0.311i)16-s + (0.965 − 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617135 + 1.27748i\)
\(L(\frac12)\) \(\approx\) \(0.617135 + 1.27748i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 + 2.21i)T \)
13 \( 1 + (-0.783 + 3.51i)T \)
good2 \( 1 + (0.273 - 1.01i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.01 - 3.80i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.669 + 0.669i)T - 11iT^{2} \)
17 \( 1 + (-3.98 + 6.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.23 - 1.23i)T - 19iT^{2} \)
23 \( 1 + (6.00 - 3.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.71 + 2.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.37 + 1.17i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-9.40 - 2.52i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.117 - 0.437i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.0936 + 0.0540i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.45 + 1.19i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.747 - 1.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.68 - 1.25i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 6.61iT - 61T^{2} \)
67 \( 1 + (-3.19 - 3.19i)T + 67iT^{2} \)
71 \( 1 + (2.68 - 10.0i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.05 + 3.95i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.473 + 0.820i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.26 + 8.26i)T - 83iT^{2} \)
89 \( 1 + (1.38 - 5.17i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-11.5 - 3.09i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.81634438362905523237601569825, −11.17001384552375157090785720820, −10.36128905469592211945578918456, −9.584857556790735294548879189271, −7.900140637606970141814898442055, −7.47679818117470605408561526469, −6.36259917771873539473264495641, −5.50386340097177474691380659162, −3.64576851770893391955049201539, −2.63878983199686812429367486949, 1.37218581419089055484453501387, 2.10235512476759414459921377135, 4.19857441203801359293451909187, 5.61015750379112014982456951015, 6.29995297417767233112942592719, 8.012165541057173137679989129575, 8.830030898548479430812003171503, 9.542129516285701492483175144643, 10.73599744826649188292353064850, 11.87367161545420385577660853943

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