Properties

Label 2-567-63.5-c1-0-22
Degree $2$
Conductor $567$
Sign $-0.959 - 0.281i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
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  − 2.23i·2-s − 3.00·4-s + (2.5 + 0.866i)7-s + 2.23i·8-s + (−3.87 − 2.23i)11-s + (−1.5 − 0.866i)13-s + (1.93 − 5.59i)14-s − 0.999·16-s + (−3.87 − 6.70i)17-s + (−3 − 1.73i)19-s + (−5.00 + 8.66i)22-s + (3.87 − 2.23i)23-s + (2.5 − 4.33i)25-s + (−1.93 + 3.35i)26-s + (−7.50 − 2.59i)28-s + (3.87 − 2.23i)29-s + ⋯
L(s)  = 1  − 1.58i·2-s − 1.50·4-s + (0.944 + 0.327i)7-s + 0.790i·8-s + (−1.16 − 0.674i)11-s + (−0.416 − 0.240i)13-s + (0.517 − 1.49i)14-s − 0.249·16-s + (−0.939 − 1.62i)17-s + (−0.688 − 0.397i)19-s + (−1.06 + 1.84i)22-s + (0.807 − 0.466i)23-s + (0.5 − 0.866i)25-s + (−0.379 + 0.657i)26-s + (−1.41 − 0.490i)28-s + (0.719 − 0.415i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.959 - 0.281i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.959 - 0.281i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156736 + 1.08977i\)
\(L(\frac12)\) \(\approx\) \(0.156736 + 1.08977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-2.5 - 0.866i)T \)
good2 \( 1 + 2.23iT - 2T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.87 + 2.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.87 + 6.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.87 + 2.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.87 + 2.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.87 - 6.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.74T + 47T^{2} \)
53 \( 1 + (-3.87 + 2.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 + 8.66iT - 61T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + 8.94iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + (-3.87 - 6.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.74 - 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \)
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

−10.69510514745588797058089419945, −9.588731475914183001103564402772, −8.743119563101379615575771887603, −7.948389739832132563264843100731, −6.62111086527781951092278588301, −5.00401695479920221390303917803, −4.60318469602394998759439561599, −2.87862213097586558618610228184, −2.38078614325983022667206964627, −0.61463765283150756453114400639, 2.06452844581411851004323445560, 4.12807895727305409895864701553, 4.97917073840321649615146251792, 5.73543586512284637792277924404, 6.98875584569393463054087728828, 7.43992375243295659487748064161, 8.433374293877543791589859228911, 8.935698750877515367880122736475, 10.43663946888804254384812168591, 10.88846897260432660514470761271

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