↑

Properties

Label 2-273-39.8-c1-0-7
Degree $2$
Conductor $273$
Sign $0.945 - 0.326i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.946 − 0.946i)2-s + (−1.73 + 0.0399i)3-s + 0.206i·4-s + (−1.18 + 1.18i)5-s + (−1.60 + 1.67i)6-s + (0.707 − 0.707i)7-s + (2.08 + 2.08i)8-s + (2.99 − 0.138i)9-s + 2.25i·10-s + (3.22 + 3.22i)11-s + (−0.00825 − 0.357i)12-s + (3.54 + 0.652i)13-s − 1.33i·14-s + (2.01 − 2.10i)15-s + 3.54·16-s − 4.20·17-s + ⋯
L(s)  = 1  + (0.669 − 0.669i)2-s + (−0.999 + 0.0230i)3-s + 0.103i·4-s + (−0.531 + 0.531i)5-s + (−0.653 + 0.684i)6-s + (0.267 − 0.267i)7-s + (0.738 + 0.738i)8-s + (0.998 − 0.0461i)9-s + 0.711i·10-s + (0.973 + 0.973i)11-s + (−0.00238 − 0.103i)12-s + (0.983 + 0.180i)13-s − 0.357i·14-s + (0.518 − 0.543i)15-s + 0.886·16-s − 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.945 - 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31071 + 0.219922i\)
\(L(\frac12)\) \(\approx\) \(1.31071 + 0.219922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.0399i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-3.54 - 0.652i)T \)
good2 \( 1 + (-0.946 + 0.946i)T - 2iT^{2} \)
5 \( 1 + (1.18 - 1.18i)T - 5iT^{2} \)
11 \( 1 + (-3.22 - 3.22i)T + 11iT^{2} \)
17 \( 1 + 4.20T + 17T^{2} \)
19 \( 1 + (-1.77 - 1.77i)T + 19iT^{2} \)
23 \( 1 + 7.43T + 23T^{2} \)
29 \( 1 - 0.211iT - 29T^{2} \)
31 \( 1 + (1.20 + 1.20i)T + 31iT^{2} \)
37 \( 1 + (-6.89 + 6.89i)T - 37iT^{2} \)
41 \( 1 + (-6.06 + 6.06i)T - 41iT^{2} \)
43 \( 1 + 8.80iT - 43T^{2} \)
47 \( 1 + (-3.55 - 3.55i)T + 47iT^{2} \)
53 \( 1 - 5.75iT - 53T^{2} \)
59 \( 1 + (7.52 + 7.52i)T + 59iT^{2} \)
61 \( 1 - 6.83T + 61T^{2} \)
67 \( 1 + (6.97 + 6.97i)T + 67iT^{2} \)
71 \( 1 + (-1.27 + 1.27i)T - 71iT^{2} \)
73 \( 1 + (3.21 - 3.21i)T - 73iT^{2} \)
79 \( 1 - 1.31T + 79T^{2} \)
83 \( 1 + (7.18 - 7.18i)T - 83iT^{2} \)
89 \( 1 + (-5.86 - 5.86i)T + 89iT^{2} \)
97 \( 1 + (-1.14 - 1.14i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.93739167618451713128602120402, −11.15842914808580879757155084056, −10.66565967667292116623149887848, −9.332581494286415411794300974540, −7.79363014149636930017352289440, −6.96696512973104510591790408238, −5.78322199520261182666136811770, −4.25577901111067058609205091599, −3.91444354307464494484215576498, −1.86353648788406611906313278257, 1.08629701151932081317389691401, 3.99328669601370707336932007061, 4.72146296500610843384628395095, 6.03642020602537282450725965625, 6.33265314925433138157238515792, 7.74944814546166395176854220793, 8.854930622429265110999003712251, 10.11345756619799639467977424373, 11.27740643745482485099885078165, 11.70488796909751662966973795151

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