Properties

Label 2-567-21.5-c1-0-10
Degree $2$
Conductor $567$
Sign $0.928 + 0.372i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.972 − 0.561i)2-s + (−0.370 − 0.640i)4-s + (−0.115 + 0.200i)5-s + (2.56 + 0.666i)7-s + 3.07i·8-s + (0.224 − 0.129i)10-s + (−0.746 + 0.430i)11-s − 2.36i·13-s + (−2.11 − 2.08i)14-s + (0.985 − 1.70i)16-s + (3.32 + 5.75i)17-s + (3.37 + 1.94i)19-s + 0.171·20-s + 0.967·22-s + (−3.05 − 1.76i)23-s + ⋯
L(s)  = 1  + (−0.687 − 0.396i)2-s + (−0.185 − 0.320i)4-s + (−0.0516 + 0.0894i)5-s + (0.967 + 0.251i)7-s + 1.08i·8-s + (0.0710 − 0.0410i)10-s + (−0.224 + 0.129i)11-s − 0.654i·13-s + (−0.565 − 0.557i)14-s + (0.246 − 0.426i)16-s + (0.806 + 1.39i)17-s + (0.773 + 0.446i)19-s + 0.0382·20-s + 0.206·22-s + (−0.637 − 0.368i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.928 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01975 - 0.196871i\)
\(L(\frac12)\) \(\approx\) \(1.01975 - 0.196871i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 0.666i)T \)
good2 \( 1 + (0.972 + 0.561i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.115 - 0.200i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.746 - 0.430i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.36iT - 13T^{2} \)
17 \( 1 + (-3.32 - 5.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.37 - 1.94i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.05 + 1.76i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.999iT - 29T^{2} \)
31 \( 1 + (-3.32 + 1.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.34 + 7.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.66T + 41T^{2} \)
43 \( 1 - 1.96T + 43T^{2} \)
47 \( 1 + (-5.32 + 9.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.45 + 4.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.43 - 7.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.12 + 2.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.82 - 8.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.77iT - 71T^{2} \)
73 \( 1 + (11.7 - 6.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.78 - 6.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + (-6.69 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.74iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.46173802729317386450147988349, −10.06401753621529191660701684701, −8.898637137606997117858663090307, −8.190720730717870788035400300788, −7.51233092669952418730182616743, −5.82768038690035545925013715527, −5.31060313951391404997185268745, −3.96899126798926533851228305517, −2.37064023246457595754852909416, −1.16162592289069407121026061505, 1.00073567312061604244093125847, 2.90489099533207048546704306516, 4.28503390918894436483097638450, 5.12893551648378508777762981283, 6.56280498467984895869789008736, 7.53548646449234485458126961253, 8.005147372639161216132683474548, 9.006099157528159634205521548458, 9.686018337277083819661426210358, 10.65014236308504445149479259897

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