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Properties

Label 2-567-63.59-c1-0-29
Degree $2$
Conductor $567$
Sign $-0.585 + 0.810i$
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  + (1.58 + 0.917i)2-s + (0.682 + 1.18i)4-s − 3.80·5-s + (−2.43 − 1.03i)7-s − 1.16i·8-s + (−6.04 − 3.49i)10-s + 1.16i·11-s + (−4.79 − 2.77i)13-s + (−2.91 − 3.88i)14-s + (2.43 − 4.21i)16-s + (−1.58 + 2.75i)17-s + (−0.546 + 0.315i)19-s + (−2.59 − 4.50i)20-s + (−1.06 + 1.85i)22-s + 1.76i·23-s + ⋯
L(s)  = 1  + (1.12 + 0.648i)2-s + (0.341 + 0.590i)4-s − 1.70·5-s + (−0.919 − 0.392i)7-s − 0.412i·8-s + (−1.91 − 1.10i)10-s + 0.351i·11-s + (−1.33 − 0.768i)13-s + (−0.778 − 1.03i)14-s + (0.608 − 1.05i)16-s + (−0.385 + 0.667i)17-s + (−0.125 + 0.0724i)19-s + (−0.580 − 1.00i)20-s + (−0.227 + 0.394i)22-s + 0.367i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.585 + 0.810i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.585 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165045 - 0.322799i\)
\(L(\frac12)\) \(\approx\) \(0.165045 - 0.322799i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.43 + 1.03i)T \)
good2 \( 1 + (-1.58 - 0.917i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 3.80T + 5T^{2} \)
11 \( 1 - 1.16iT - 11T^{2} \)
13 \( 1 + (4.79 + 2.77i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.58 - 2.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.546 - 0.315i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.76iT - 23T^{2} \)
29 \( 1 + (4.18 - 2.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.70 + 2.13i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.28 - 3.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.61 + 6.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.27 + 2.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0627 + 0.0362i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.49 - 6.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.00 + 4.04i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.54 + 4.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.76iT - 71T^{2} \)
73 \( 1 + (-4.84 - 2.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.54 - 14.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.08 + 8.80i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.77 + 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.596 + 0.344i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.48328072792981285228016827629, −9.665028370082568735005088278550, −8.321544227754584569375604525134, −7.30070675903469871321781760790, −7.01892193760923576078162933848, −5.73330867115301140951194798744, −4.64206851123637312400400505719, −3.91151616279985044855084370435, −3.08003382576445578462889974781, −0.13592154980079847053344395361, 2.55090569672864682119221460254, 3.39771244303281068919354911708, 4.32337177434515218491087992932, 5.04528022063977377938487753588, 6.46297218195887717576087705719, 7.38798891574116154160430718797, 8.380271242429913558764986245851, 9.326631899281437415644991147241, 10.52211801106671006296144466283, 11.59559176339705614317299647207

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