↑

Properties

Label 2-273-273.254-c1-0-9
Degree $2$
Conductor $273$
Sign $0.988 + 0.148i$
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  + (−1.12 − 0.301i)2-s + (−0.956 + 1.44i)3-s + (−0.555 − 0.320i)4-s + (−0.0307 − 0.114i)5-s + (1.51 − 1.33i)6-s + (−1.83 − 1.90i)7-s + (2.17 + 2.17i)8-s + (−1.16 − 2.76i)9-s + 0.138i·10-s + (2.30 + 2.30i)11-s + (0.995 − 0.495i)12-s + (2.91 − 2.12i)13-s + (1.48 + 2.70i)14-s + (0.194 + 0.0653i)15-s + (−1.15 − 1.99i)16-s + (0.665 − 1.15i)17-s + ⋯
L(s)  = 1  + (−0.795 − 0.213i)2-s + (−0.552 + 0.833i)3-s + (−0.277 − 0.160i)4-s + (−0.0137 − 0.0512i)5-s + (0.617 − 0.545i)6-s + (−0.692 − 0.721i)7-s + (0.769 + 0.769i)8-s + (−0.389 − 0.920i)9-s + 0.0437i·10-s + (0.695 + 0.695i)11-s + (0.287 − 0.143i)12-s + (0.808 − 0.589i)13-s + (0.397 + 0.722i)14-s + (0.0503 + 0.0168i)15-s + (−0.288 − 0.498i)16-s + (0.161 − 0.279i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.988 + 0.148i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612622 - 0.0456210i\)
\(L(\frac12)\) \(\approx\) \(0.612622 - 0.0456210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.956 - 1.44i)T \)
7 \( 1 + (1.83 + 1.90i)T \)
13 \( 1 + (-2.91 + 2.12i)T \)
good2 \( 1 + (1.12 + 0.301i)T + (1.73 + i)T^{2} \)
5 \( 1 + (0.0307 + 0.114i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.30 - 2.30i)T + 11iT^{2} \)
17 \( 1 + (-0.665 + 1.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.29 - 3.29i)T + 19iT^{2} \)
23 \( 1 + (-1.93 - 3.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.77 - 2.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.08 + 7.79i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.91 - 0.512i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.06 + 11.4i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.721 - 0.416i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.90 - 1.85i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.67 - 3.27i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.47 + 1.19i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 8.26T + 61T^{2} \)
67 \( 1 + (0.646 + 0.646i)T + 67iT^{2} \)
71 \( 1 + (-15.9 - 4.26i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (12.6 + 3.38i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.52 + 2.64i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.83 - 2.83i)T - 83iT^{2} \)
89 \( 1 + (1.62 - 6.05i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.43 + 12.8i)T + (-84.0 - 48.5i)T^{2} \)
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

−11.54019545815018559124053401683, −10.64281914402703671888322755646, −9.923742578661035039156689651747, −9.400229123972569200568172344136, −8.310351105394360010973760522769, −6.99638387418885343566704329109, −5.76505355222807521093180658776, −4.59657841697579108034127734936, −3.49709308175194693248198796064, −0.929861588861227079151759657761, 1.09063437366182373252747475895, 3.18432306891215279991246160962, 4.92471563108568361250550713620, 6.38968175386796842410675151208, 6.84566895550873805081337203026, 8.307554669546901856668155664784, 8.800135007997367822940122241922, 9.854195337634118167877701210199, 11.06428798074744591019912612892, 11.87039412778382291087567730494

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