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Properties

Label 2-273-91.9-c1-0-0
Degree $2$
Conductor $273$
Sign $0.264 - 0.964i$
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  + (−1.14 − 1.97i)2-s + 3-s + (−1.59 + 2.77i)4-s + (−1.46 + 2.54i)5-s + (−1.14 − 1.97i)6-s + (−2.34 − 1.23i)7-s + 2.73·8-s + 9-s + 6.69·10-s − 5.16·11-s + (−1.59 + 2.77i)12-s + (−0.364 + 3.58i)13-s + (0.233 + 6.02i)14-s + (−1.46 + 2.54i)15-s + (0.0801 + 0.138i)16-s + (−2.52 + 4.37i)17-s + ⋯
L(s)  = 1  + (−0.806 − 1.39i)2-s + 0.577·3-s + (−0.799 + 1.38i)4-s + (−0.656 + 1.13i)5-s + (−0.465 − 0.806i)6-s + (−0.884 − 0.466i)7-s + 0.967·8-s + 0.333·9-s + 2.11·10-s − 1.55·11-s + (−0.461 + 0.799i)12-s + (−0.101 + 0.994i)13-s + (0.0623 + 1.61i)14-s + (−0.379 + 0.656i)15-s + (0.0200 + 0.0346i)16-s + (−0.612 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.264 - 0.964i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.264 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211317 + 0.161200i\)
\(L(\frac12)\) \(\approx\) \(0.211317 + 0.161200i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.34 + 1.23i)T \)
13 \( 1 + (0.364 - 3.58i)T \)
good2 \( 1 + (1.14 + 1.97i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.46 - 2.54i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 5.16T + 11T^{2} \)
17 \( 1 + (2.52 - 4.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 2.25T + 19T^{2} \)
23 \( 1 + (2.61 + 4.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.216 + 0.375i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.34 + 2.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.12 - 3.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.269 + 0.466i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.66 + 8.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.87 - 8.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.377 - 0.653i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.82 + 3.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.95T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + (-3.90 - 6.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.94 + 13.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.79 - 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (-2.00 - 3.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.69 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.85499954146439601763522453592, −10.82073986998749469808775643715, −10.39466151340091891362960485244, −9.571215472332122688224345738844, −8.425874402897910428392454623068, −7.56207085638502409758802766353, −6.47337269330617863778273850089, −4.13196748912664376201276798796, −3.18480363652634956151858695431, −2.29295608409075427490418677702, 0.23134820366535718720876275752, 3.03886051207794077392986982100, 4.94067627904205450768170680945, 5.69258011354530234820583499392, 7.18434310085678744688545653654, 7.905979329545976449562056272339, 8.582454810376646569978990130976, 9.420998898914046431245822993544, 10.17950275770757853885900103820, 11.85333237193913508344220505323

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