Properties

Label 2-273-273.185-c1-0-26
Degree $2$
Conductor $273$
Sign $-0.0850 + 0.996i$
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  + (−0.0436 − 0.0252i)2-s + (1.71 − 0.221i)3-s + (−0.998 − 1.72i)4-s + (−1.19 − 2.07i)5-s + (−0.0806 − 0.0336i)6-s + (−2.29 + 1.31i)7-s + 0.201i·8-s + (2.90 − 0.761i)9-s + 0.120i·10-s − 3.29i·11-s + (−2.09 − 2.75i)12-s + (3.26 − 1.53i)13-s + (0.133 + 0.000331i)14-s + (−2.51 − 3.29i)15-s + (−1.99 + 3.45i)16-s + (−3.17 − 5.49i)17-s + ⋯
L(s)  = 1  + (−0.0308 − 0.0178i)2-s + (0.991 − 0.127i)3-s + (−0.499 − 0.864i)4-s + (−0.534 − 0.926i)5-s + (−0.0329 − 0.0137i)6-s + (−0.867 + 0.497i)7-s + 0.0712i·8-s + (0.967 − 0.253i)9-s + 0.0381i·10-s − 0.994i·11-s + (−0.605 − 0.793i)12-s + (0.904 − 0.427i)13-s + (0.0356 + 8.84e−5i)14-s + (−0.648 − 0.850i)15-s + (−0.498 + 0.862i)16-s + (−0.770 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0850 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0850 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0850 + 0.996i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0850 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869408 - 0.946809i\)
\(L(\frac12)\) \(\approx\) \(0.869408 - 0.946809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.221i)T \)
7 \( 1 + (2.29 - 1.31i)T \)
13 \( 1 + (-3.26 + 1.53i)T \)
good2 \( 1 + (0.0436 + 0.0252i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.19 + 2.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.29iT - 11T^{2} \)
17 \( 1 + (3.17 + 5.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 4.97iT - 19T^{2} \)
23 \( 1 + (-3.28 - 1.89i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.15 + 1.24i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.40 - 2.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.62 - 4.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.726 - 1.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.219 - 0.379i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.16 - 3.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.7 - 6.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.41 + 2.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.84iT - 61T^{2} \)
67 \( 1 - 1.43T + 67T^{2} \)
71 \( 1 + (5.14 + 2.97i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.79 - 2.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.32 - 4.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + (-2.54 + 4.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.0 + 6.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.78333166052945698993951104122, −10.48623063247651102355049921837, −9.470589799792539430300977200630, −8.780462298571675024477578949540, −8.226606427367968201639825004774, −6.63986797794756864911589597317, −5.51495314586148297523830588933, −4.25798987498415706569401643978, −3.01554189926912685641022262245, −0.980367932946824946925581546785, 2.62701045837430512199899092367, 3.71563884079956728331681327655, 4.34311666903122117187128446465, 6.80834622474674098215680655673, 7.15983096650268171128292897117, 8.406224322284997932013919638230, 9.105612076640127506700886599306, 10.18811645034353438632983139833, 11.06826497174119422333801542293, 12.41800967225461452409640670085

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