Properties

Label 2-273-91.80-c1-0-4
Degree $2$
Conductor $273$
Sign $-0.534 - 0.845i$
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  + (0.170 + 0.636i)2-s + i·3-s + (1.35 − 0.782i)4-s + (−1.02 + 3.81i)5-s + (−0.636 + 0.170i)6-s + (−2.47 + 0.938i)7-s + (1.66 + 1.66i)8-s − 9-s − 2.60·10-s + (−1.90 − 1.90i)11-s + (0.782 + 1.35i)12-s + (−0.360 − 3.58i)13-s + (−1.01 − 1.41i)14-s + (−3.81 − 1.02i)15-s + (0.791 − 1.37i)16-s + (1.67 + 2.89i)17-s + ⋯
L(s)  = 1  + (0.120 + 0.450i)2-s + 0.577i·3-s + (0.678 − 0.391i)4-s + (−0.457 + 1.70i)5-s + (−0.259 + 0.0696i)6-s + (−0.935 + 0.354i)7-s + (0.587 + 0.587i)8-s − 0.333·9-s − 0.823·10-s + (−0.573 − 0.573i)11-s + (0.226 + 0.391i)12-s + (−0.0999 − 0.994i)13-s + (−0.272 − 0.378i)14-s + (−0.986 − 0.264i)15-s + (0.197 − 0.342i)16-s + (0.405 + 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.534 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.534 - 0.845i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.534 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.631255 + 1.14556i\)
\(L(\frac12)\) \(\approx\) \(0.631255 + 1.14556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (2.47 - 0.938i)T \)
13 \( 1 + (0.360 + 3.58i)T \)
good2 \( 1 + (-0.170 - 0.636i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (1.02 - 3.81i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.90 + 1.90i)T + 11iT^{2} \)
17 \( 1 + (-1.67 - 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.69 - 4.69i)T + 19iT^{2} \)
23 \( 1 + (-5.65 - 3.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.95 + 3.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.41 + 0.378i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.02 - 0.542i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.717 - 2.67i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9.41 - 5.43i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.76 - 0.473i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.90 + 6.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.574 + 0.153i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 5.48iT - 61T^{2} \)
67 \( 1 + (2.33 - 2.33i)T - 67iT^{2} \)
71 \( 1 + (0.629 + 2.35i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.670 + 2.50i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.72 + 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.485 + 0.485i)T + 83iT^{2} \)
89 \( 1 + (2.12 + 7.93i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.37 + 1.97i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.94417565663380955800789691482, −11.04985822289964363398895162370, −10.41578590640806951518402348870, −9.762492350829960399137982594792, −7.995570674956008791199048139077, −7.28821791093787661008795543361, −6.13056706823138896482006039636, −5.57235812600971549928766094442, −3.43550045545556069086252705432, −2.83237218638771092067755471143, 1.01043778840463875219088147689, 2.71685566839330695287359603630, 4.17037744769941269055841474326, 5.29820303384730511237041704836, 6.99915770597653236128635037112, 7.42757216764030978280158267120, 8.804611800652102332058482575806, 9.536640057530451174305896876904, 10.90144292069514148469358323072, 11.97436450550489052150022610387

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