↑

Properties

Label 2-273-13.12-c1-0-14
Degree $2$
Conductor $273$
Sign $-0.832 + 0.554i$
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  − 2i·2-s + 3-s − 2·4-s − 3i·5-s − 2i·6-s + i·7-s + 9-s − 6·10-s − 2·12-s + (2 + 3i)13-s + 2·14-s − 3i·15-s − 4·16-s − 2·17-s − 2i·18-s − i·19-s + ⋯
L(s)  = 1  − 1.41i·2-s + 0.577·3-s − 4-s − 1.34i·5-s − 0.816i·6-s + 0.377i·7-s + 0.333·9-s − 1.89·10-s − 0.577·12-s + (0.554 + 0.832i)13-s + 0.534·14-s − 0.774i·15-s − 16-s − 0.485·17-s − 0.471i·18-s − 0.229i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.434397 - 1.43471i\)
\(L(\frac12)\) \(\approx\) \(0.434397 - 1.43471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - iT \)
13 \( 1 + (-2 - 3i)T \)
good2 \( 1 + 2iT - 2T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 5iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 9iT - 73T^{2} \)
79 \( 1 - 15T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 - 13iT - 97T^{2} \)
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

−11.75942999023472771407128015184, −10.65135977649544478028872182113, −9.495803727925246431264251006851, −9.012452303660284311020256051581, −8.132154481969938936747522698136, −6.47642883402419071704485276541, −4.81595767110589909730428898467, −3.98036133172615340916797878280, −2.51396560601296405833713333615, −1.25303551678348609773086333816, 2.65376140931775816657558005927, 4.00210781752750486635416871863, 5.58698705029836725130809403607, 6.60369478835799314994114036829, 7.29523439370080668495912550963, 8.106523840292540468554590842384, 9.091325888207403476248922949636, 10.44237773289844884506708290251, 11.00019349193892584344335973713, 12.53711924879391049248068412288

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