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Properties

Label 2-273-91.24-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.747 - 0.664i$
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  + (0.562 − 0.150i)2-s + i·3-s + (−1.43 + 0.830i)4-s + (−0.672 − 0.180i)5-s + (0.150 + 0.562i)6-s + (−2.49 + 0.881i)7-s + (−1.50 + 1.50i)8-s − 9-s − 0.405·10-s + (−0.628 + 0.628i)11-s + (−0.830 − 1.43i)12-s + (−0.659 + 3.54i)13-s + (−1.27 + 0.871i)14-s + (0.180 − 0.672i)15-s + (1.04 − 1.80i)16-s + (−0.0230 − 0.0398i)17-s + ⋯
L(s)  = 1  + (0.397 − 0.106i)2-s + 0.577i·3-s + (−0.719 + 0.415i)4-s + (−0.300 − 0.0806i)5-s + (0.0615 + 0.229i)6-s + (−0.942 + 0.333i)7-s + (−0.532 + 0.532i)8-s − 0.333·9-s − 0.128·10-s + (−0.189 + 0.189i)11-s + (−0.239 − 0.415i)12-s + (−0.183 + 0.983i)13-s + (−0.339 + 0.233i)14-s + (0.0465 − 0.173i)15-s + (0.260 − 0.450i)16-s + (−0.00558 − 0.00967i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.747 - 0.664i$
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: \((2,\ 273,\ (\ :1/2),\ -0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264169 + 0.695275i\)
\(L(\frac12)\) \(\approx\) \(0.264169 + 0.695275i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 - 0.881i)T \)
13 \( 1 + (0.659 - 3.54i)T \)
good2 \( 1 + (-0.562 + 0.150i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.672 + 0.180i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.628 - 0.628i)T - 11iT^{2} \)
17 \( 1 + (0.0230 + 0.0398i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.617 - 0.617i)T - 19iT^{2} \)
23 \( 1 + (-2.76 - 1.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.08 - 7.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.04 - 3.90i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.82 + 6.80i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-8.56 - 2.29i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.29 - 0.746i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.24 - 12.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.89 + 8.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.28 - 8.54i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 1.01iT - 61T^{2} \)
67 \( 1 + (1.77 + 1.77i)T + 67iT^{2} \)
71 \( 1 + (0.798 - 0.213i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (15.2 - 4.09i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.73 + 8.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.50 + 3.50i)T - 83iT^{2} \)
89 \( 1 + (5.05 - 1.35i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.288 - 1.07i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.36177188442917942663746327017, −11.52949909376704245792809249959, −10.27843485373274293379519203610, −9.278836501289353515829345714997, −8.756337726174597937330225261827, −7.42080179642200340153693312986, −6.08256768140572994695828451707, −4.87943334703888441351484234623, −3.95229336055989948566255476761, −2.86255088088278823365540121759, 0.50325640476973949788099493548, 2.96392066171204046058071266316, 4.18845488277339413586404687322, 5.55669632199699229990612995637, 6.40026763212071030517942512734, 7.55500164900318081998618558635, 8.603196321444544228448171494349, 9.728369910077853738127685909064, 10.46390684208963469319707586911, 11.76964439163454932612692172953

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