↑

Properties

Label 2-567-9.5-c2-0-18
Degree $2$
Conductor $567$
Sign $0.984 - 0.173i$
Analytic cond. $15.4496$
Root an. cond. $3.93060$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.625 − 0.361i)2-s + (−1.73 + 3.01i)4-s + (−6.88 − 3.97i)5-s + (−1.32 − 2.29i)7-s + 5.40i·8-s − 5.74·10-s + (−8.01 + 4.62i)11-s + (11.0 − 19.1i)13-s + (−1.65 − 0.955i)14-s + (−5.00 − 8.66i)16-s + 27.2i·17-s + 31.8·19-s + (23.9 − 13.8i)20-s + (−3.34 + 5.79i)22-s + (11.7 + 6.76i)23-s + ⋯
L(s)  = 1  + (0.312 − 0.180i)2-s + (−0.434 + 0.752i)4-s + (−1.37 − 0.794i)5-s + (−0.188 − 0.327i)7-s + 0.675i·8-s − 0.574·10-s + (−0.728 + 0.420i)11-s + (0.851 − 1.47i)13-s + (−0.118 − 0.0682i)14-s + (−0.312 − 0.541i)16-s + 1.60i·17-s + 1.67·19-s + (1.19 − 0.690i)20-s + (−0.151 + 0.263i)22-s + (0.509 + 0.294i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(15.4496\)
Root analytic conductor: \(3.93060\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.267301174\)
\(L(\frac12)\) \(\approx\) \(1.267301174\)
\(L(2)\) 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 \( 1 + (1.32 + 2.29i)T \)
good2 \( 1 + (-0.625 + 0.361i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (6.88 + 3.97i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.01 - 4.62i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-11.0 + 19.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 27.2iT - 289T^{2} \)
19 \( 1 - 31.8T + 361T^{2} \)
23 \( 1 + (-11.7 - 6.76i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-5.83 + 3.36i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (5.96 - 10.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 26.8T + 1.36e3T^{2} \)
41 \( 1 + (-55.3 - 31.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-8.10 - 14.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (0.448 - 0.259i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 71.8iT - 2.80e3T^{2} \)
59 \( 1 + (-33.3 - 19.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (13.9 + 24.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (14.7 - 25.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 39.4iT - 5.04e3T^{2} \)
73 \( 1 - 67.1T + 5.32e3T^{2} \)
79 \( 1 + (-9.38 - 16.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (69.3 - 40.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 62.9iT - 7.92e3T^{2} \)
97 \( 1 + (-23.3 - 40.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−10.81026307184145384918480742303, −9.636745490612768466235213932341, −8.433072760513815147950553833309, −8.010953757227315760486093996278, −7.39888941913231575318992982798, −5.66648733625795879373257396474, −4.74042238394651113952757437173, −3.78125931426126126277957880351, −3.12575289052457453711382983987, −0.838458968417780090437524057696, 0.69675955287730584246755505679, 2.82704427218436926645540158152, 3.87489957287842003725040782415, 4.84516179657410929270157185575, 5.90860532592583966649453205452, 6.96962152956409334907142975510, 7.59298728863725550614550092453, 8.895406557705919655543264086707, 9.522282403741556673270219570735, 10.72621775995218321497704547310

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