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Properties

Label 2-273-91.59-c1-0-18
Degree $2$
Conductor $273$
Sign $-0.431 - 0.902i$
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  + (−1.22 − 1.22i)2-s + (0.866 − 0.5i)3-s + 1.02i·4-s + (−1.95 − 0.524i)5-s + (−1.67 − 0.449i)6-s + (−1.82 + 1.92i)7-s + (−1.20 + 1.20i)8-s + (0.499 − 0.866i)9-s + (1.76 + 3.05i)10-s + (−3.50 − 0.938i)11-s + (0.511 + 0.885i)12-s + (−1.78 + 3.13i)13-s + (4.59 − 0.123i)14-s + (−1.95 + 0.524i)15-s + 4.99·16-s − 1.50·17-s + ⋯
L(s)  = 1  + (−0.869 − 0.869i)2-s + (0.499 − 0.288i)3-s + 0.511i·4-s + (−0.875 − 0.234i)5-s + (−0.685 − 0.183i)6-s + (−0.687 + 0.725i)7-s + (−0.424 + 0.424i)8-s + (0.166 − 0.288i)9-s + (0.556 + 0.964i)10-s + (−1.05 − 0.282i)11-s + (0.147 + 0.255i)12-s + (−0.496 + 0.868i)13-s + (1.22 − 0.0328i)14-s + (−0.505 + 0.135i)15-s + 1.24·16-s − 0.365·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.431 - 0.902i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.431 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0346451 + 0.0549476i\)
\(L(\frac12)\) \(\approx\) \(0.0346451 + 0.0549476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (1.82 - 1.92i)T \)
13 \( 1 + (1.78 - 3.13i)T \)
good2 \( 1 + (1.22 + 1.22i)T + 2iT^{2} \)
5 \( 1 + (1.95 + 0.524i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.50 + 0.938i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + (1.07 + 3.99i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 3.71iT - 23T^{2} \)
29 \( 1 + (1.84 - 3.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.89 - 7.05i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.85 + 1.85i)T - 37iT^{2} \)
41 \( 1 + (1.33 + 4.97i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.51 - 2.60i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0684 + 0.255i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.63 - 4.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.912 - 0.912i)T + 59iT^{2} \)
61 \( 1 + (9.19 + 5.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.47 + 12.9i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.11 + 11.6i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.77 - 0.744i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.09 + 14.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.3 - 10.3i)T - 83iT^{2} \)
89 \( 1 + (-7.11 - 7.11i)T + 89iT^{2} \)
97 \( 1 + (0.645 + 0.173i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.20682414600050673309404858930, −10.30589403627070328396230704067, −9.179362295874296594759800535252, −8.706610520458006179078041319459, −7.72553940429404996031580806661, −6.46420593421485022281886651354, −4.89707505875763272288233564025, −3.20777279438795672050287961178, −2.21913034365176255783469268904, −0.05770882027911960509757836864, 3.04585961357718623292603609099, 4.11438598964697923485524792060, 5.80819970956113551345348299188, 7.17681221828975507117732216807, 7.73495735351038669540644770390, 8.362141500077479040221590080521, 9.831019474630209336244212171096, 10.07758139820559528446760142984, 11.42357186713349067018486012849, 12.72060189273642917528031489812

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