↑

Properties

Label 2-279-31.2-c1-0-5
Degree $2$
Conductor $279$
Sign $0.606 - 0.795i$
Analytic cond. $2.22782$
Root an. cond. $1.49259$
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  + (1.30 + 0.951i)2-s + (0.190 + 0.587i)4-s + 0.381·5-s + (0.927 + 2.85i)7-s + (0.690 − 2.12i)8-s + (0.5 + 0.363i)10-s + (1.61 + 4.97i)11-s + (1.5 − 1.08i)13-s + (−1.5 + 4.61i)14-s + (3.92 − 2.85i)16-s + (1.30 − 4.02i)17-s + (−4.04 − 2.93i)19-s + (0.0729 + 0.224i)20-s + (−2.61 + 8.05i)22-s + (−1.07 + 3.30i)23-s + ⋯
L(s)  = 1  + (0.925 + 0.672i)2-s + (0.0954 + 0.293i)4-s + 0.170·5-s + (0.350 + 1.07i)7-s + (0.244 − 0.751i)8-s + (0.158 + 0.114i)10-s + (0.487 + 1.50i)11-s + (0.416 − 0.302i)13-s + (−0.400 + 1.23i)14-s + (0.981 − 0.713i)16-s + (0.317 − 0.977i)17-s + (−0.928 − 0.674i)19-s + (0.0163 + 0.0502i)20-s + (−0.558 + 1.71i)22-s + (−0.223 + 0.688i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(279\)    =    \(3^{2} \cdot 31\)
Sign: $0.606 - 0.795i$
Analytic conductor: \(2.22782\)
Root analytic conductor: \(1.49259\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{279} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 279,\ (\ :1/2),\ 0.606 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88652 + 0.934163i\)
\(L(\frac12)\) \(\approx\) \(1.88652 + 0.934163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (-0.0450 + 5.56i)T \)
good2 \( 1 + (-1.30 - 0.951i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 - 0.381T + 5T^{2} \)
7 \( 1 + (-0.927 - 2.85i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.61 - 4.97i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.5 + 1.08i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.30 + 4.02i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.04 + 2.93i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.07 - 3.30i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.16 + 3.75i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 + (2 + 1.45i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.92 - 1.40i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (4.54 - 3.30i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.218 - 0.673i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.427 + 0.310i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 0.236T + 67T^{2} \)
71 \( 1 + (-3.42 + 10.5i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.57 - 10.9i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.73 - 4.16i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.66 - 8.19i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.78 + 17.7i)T + (-78.4 + 57.0i)T^{2} \)
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

−12.20840174558528842629300956253, −11.37623187455122509793144899284, −9.879033045507711991072079022160, −9.283094760523699195684158089947, −7.86039253425187844213349605585, −6.86597675769365490336632553962, −5.83403110216651723398086612634, −5.02757789045574184229242911871, −3.95722764448767365818005539288, −2.12261340462525093303423541846, 1.66819798200942430123061711721, 3.53530848668446276885325422941, 4.04984453500570552549329939469, 5.47691770592816077661928468607, 6.51076936574072285065087668754, 8.018009943306424607751923710938, 8.701627129166881505870307909436, 10.38317343616921149674895060129, 10.89244317522236823944657906825, 11.74536640179588118061957752662

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