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Properties

Label 2-273-91.88-c1-0-1
Degree $2$
Conductor $273$
Sign $0.229 - 0.973i$
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  + (−0.485 − 0.280i)2-s − 3-s + (−0.842 − 1.45i)4-s + (−0.449 + 0.259i)5-s + (0.485 + 0.280i)6-s + (−1.87 + 1.86i)7-s + 2.06i·8-s + 9-s + 0.291·10-s + 2.16i·11-s + (0.842 + 1.45i)12-s + (3.29 + 1.45i)13-s + (1.43 − 0.381i)14-s + (0.449 − 0.259i)15-s + (−1.10 + 1.91i)16-s + (2.48 + 4.29i)17-s + ⋯
L(s)  = 1  + (−0.343 − 0.198i)2-s − 0.577·3-s + (−0.421 − 0.729i)4-s + (−0.201 + 0.116i)5-s + (0.198 + 0.114i)6-s + (−0.708 + 0.705i)7-s + 0.730i·8-s + 0.333·9-s + 0.0921·10-s + 0.653i·11-s + (0.243 + 0.421i)12-s + (0.915 + 0.403i)13-s + (0.383 − 0.101i)14-s + (0.116 − 0.0670i)15-s + (−0.276 + 0.478i)16-s + (0.601 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399357 + 0.316189i\)
\(L(\frac12)\) \(\approx\) \(0.399357 + 0.316189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + (1.87 - 1.86i)T \)
13 \( 1 + (-3.29 - 1.45i)T \)
good2 \( 1 + (0.485 + 0.280i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.449 - 0.259i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.16iT - 11T^{2} \)
17 \( 1 + (-2.48 - 4.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 5.52iT - 19T^{2} \)
23 \( 1 + (3.39 - 5.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.81 - 6.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.15 + 2.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.06 + 2.92i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.23 - 5.33i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.11 - 3.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.69 + 5.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.04 + 1.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.63 - 1.52i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.24T + 61T^{2} \)
67 \( 1 - 1.98iT - 67T^{2} \)
71 \( 1 + (8.61 + 4.97i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.47 - 1.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.48 - 4.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.65iT - 83T^{2} \)
89 \( 1 + (2.19 + 1.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.01 - 4.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.91662640549258322126179160938, −11.10523656289448832305540071392, −10.18568979681111191268683447348, −9.376519344913706762832008201442, −8.553338178188727026967689472423, −7.05295270457036192369067424504, −5.98268034291312409902121598777, −5.17999484969169315637670619411, −3.69267829975266567368830813811, −1.69518071781820973695474564180, 0.48813170787892359512145065553, 3.34795910699591094787215252637, 4.23579858428362268846456952551, 5.80122674447722540463392110922, 6.81843208812574804912033945578, 7.888932811942511396719142185979, 8.657067016070253280471071822147, 9.939113037699122105419910583877, 10.54304251225494352852598842125, 11.95191163267226443163863808517

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