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Properties

Label 2-567-9.4-c1-0-9
Degree $2$
Conductor $567$
Sign $0.766 - 0.642i$
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  + (0.261 + 0.453i)2-s + (0.862 − 1.49i)4-s + (−1.10 + 1.90i)5-s + (−0.5 − 0.866i)7-s + 1.95·8-s − 1.15·10-s + (2.60 + 4.50i)11-s + (−1.57 + 2.73i)13-s + (0.261 − 0.453i)14-s + (−1.21 − 2.10i)16-s + 3.24·17-s + 7.45·19-s + (1.89 + 3.28i)20-s + (−1.36 + 2.36i)22-s + (2.20 − 3.81i)23-s + ⋯
L(s)  = 1  + (0.185 + 0.320i)2-s + (0.431 − 0.747i)4-s + (−0.492 + 0.852i)5-s + (−0.188 − 0.327i)7-s + 0.690·8-s − 0.364·10-s + (0.784 + 1.35i)11-s + (−0.437 + 0.757i)13-s + (0.0700 − 0.121i)14-s + (−0.303 − 0.525i)16-s + 0.788·17-s + 1.70·19-s + (0.424 + 0.735i)20-s + (−0.290 + 0.503i)22-s + (0.459 − 0.795i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63632 + 0.595572i\)
\(L(\frac12)\) \(\approx\) \(1.63632 + 0.595572i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.261 - 0.453i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.10 - 1.90i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.60 - 4.50i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.57 - 2.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.24T + 17T^{2} \)
19 \( 1 - 7.45T + 19T^{2} \)
23 \( 1 + (-2.20 + 3.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.576 + 0.998i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (5.72 - 9.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.64 + 8.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.523 + 0.907i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.249T + 53T^{2} \)
59 \( 1 + (4.04 - 7.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.30 + 7.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.80 + 6.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.60T + 71T^{2} \)
73 \( 1 - 0.846T + 73T^{2} \)
79 \( 1 + (-3.80 - 6.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.72 - 9.91i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.24T + 89T^{2} \)
97 \( 1 + (-1.72 - 2.98i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.84390503092126121788729837835, −9.880111694415850998196346558416, −9.477582679775406040159812384596, −7.76522888883317172337120450283, −7.00241980219147434767952021006, −6.66447675804929901920278832007, −5.28597662720470994052125869723, −4.32528498169946433259902604946, −3.02344082744635252027399671844, −1.51980847526357409828092862039, 1.12052124353740472277859379283, 3.03880509645944097490488191831, 3.63064374525355337760403192760, 4.99692377705375041502021025481, 5.93176558702668196383645495398, 7.29040501117436660263236674566, 8.001082040282147067018779959326, 8.793093429375194052361606399573, 9.702560494530189758093553926273, 10.92957844091558170170554271331

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