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Properties

Label 2-273-91.73-c1-0-15
Degree $2$
Conductor $273$
Sign $-0.764 + 0.645i$
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  + (0.0608 − 0.227i)2-s + (−0.866 − 0.5i)3-s + (1.68 + 0.972i)4-s + (−3.32 − 0.890i)5-s + (−0.166 + 0.166i)6-s + (−2.22 − 1.43i)7-s + (0.655 − 0.655i)8-s + (0.499 + 0.866i)9-s + (−0.404 + 0.700i)10-s + (−1.42 − 5.31i)11-s + (−0.972 − 1.68i)12-s + (−3.57 − 0.458i)13-s + (−0.461 + 0.416i)14-s + (2.43 + 2.43i)15-s + (1.83 + 3.17i)16-s + (2.51 − 4.35i)17-s + ⋯
L(s)  = 1  + (0.0430 − 0.160i)2-s + (−0.499 − 0.288i)3-s + (0.842 + 0.486i)4-s + (−1.48 − 0.398i)5-s + (−0.0678 + 0.0678i)6-s + (−0.839 − 0.543i)7-s + (0.231 − 0.231i)8-s + (0.166 + 0.288i)9-s + (−0.127 + 0.221i)10-s + (−0.429 − 1.60i)11-s + (−0.280 − 0.486i)12-s + (−0.991 − 0.127i)13-s + (−0.123 + 0.111i)14-s + (0.628 + 0.628i)15-s + (0.458 + 0.794i)16-s + (0.609 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.764 + 0.645i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.764 + 0.645i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.199236 - 0.544919i\)
\(L(\frac12)\) \(\approx\) \(0.199236 - 0.544919i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 1.43i)T \)
13 \( 1 + (3.57 + 0.458i)T \)
good2 \( 1 + (-0.0608 + 0.227i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (3.32 + 0.890i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.42 + 5.31i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.51 + 4.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.716 - 0.191i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (6.29 - 3.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.36T + 29T^{2} \)
31 \( 1 + (0.332 + 1.23i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.422 - 0.113i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.07 - 3.07i)T - 41iT^{2} \)
43 \( 1 + 9.87iT - 43T^{2} \)
47 \( 1 + (1.03 - 3.85i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.901 - 1.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.05 + 1.89i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.78 + 5.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-14.3 + 3.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.61 + 1.61i)T + 71iT^{2} \)
73 \( 1 + (2.63 - 0.704i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.66 + 6.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.05 - 1.05i)T - 83iT^{2} \)
89 \( 1 + (-0.370 + 1.38i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.29 - 3.29i)T - 97iT^{2} \)
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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

−11.72979770025375172734554378609, −10.92588275038592059695443198489, −9.861823866820126928924159699708, −8.220162861467033261227338322587, −7.62500534268674778103143817117, −6.78995118404996951961679938991, −5.48067621418984896674642107148, −3.90137991120458015535502215699, −3.00773188223000231534268259533, −0.43579888546684006798765685596, 2.46531988536334024989147839046, 3.96894480404207062882127276699, 5.19274088709201508486764905374, 6.48775666602314897643040705029, 7.21224031738851358427350094569, 8.121785954243453173274325818650, 9.935209995380919800770988287248, 10.22914860973856506185471940154, 11.48201806271153294158131189419, 12.19717640667706908109718044635

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