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Properties

Label 2-567-63.59-c1-0-20
Degree $2$
Conductor $567$
Sign $0.916 + 0.400i$
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  + (1.22 + 0.707i)2-s − 2.44·5-s + (2.5 − 0.866i)7-s − 2.82i·8-s + (−2.99 − 1.73i)10-s − 1.41i·11-s + (4.5 + 2.59i)13-s + (3.67 + 0.707i)14-s + (2.00 − 3.46i)16-s + (2.44 − 4.24i)17-s + (1.5 − 0.866i)19-s + (1.00 − 1.73i)22-s − 5.65i·23-s + 0.999·25-s + (3.67 + 6.36i)26-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s − 1.09·5-s + (0.944 − 0.327i)7-s − 0.999i·8-s + (−0.948 − 0.547i)10-s − 0.426i·11-s + (1.24 + 0.720i)13-s + (0.981 + 0.188i)14-s + (0.500 − 0.866i)16-s + (0.594 − 1.02i)17-s + (0.344 − 0.198i)19-s + (0.213 − 0.369i)22-s − 1.17i·23-s + 0.199·25-s + (0.720 + 1.24i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.916 + 0.400i$
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: \((2,\ 567,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97103 - 0.412328i\)
\(L(\frac12)\) \(\approx\) \(1.97103 - 0.412328i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 0.707i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.44T + 5T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.44 + 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 + (-2.44 + 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.67 - 6.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.12 - 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.44 + 1.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.44 - 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.67 - 6.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.44 + 4.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9 - 5.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.03088147010024483327009036040, −9.841986075784309960037952087325, −8.680892339168043670670756714350, −7.893871690998630528431346819007, −6.98575851100083715731601629340, −6.06503562516872818945650245285, −4.87631840434462117101426677532, −4.27078655163865553323861757996, −3.28164616695823343451397348105, −1.02025702418944601500794049073, 1.71545602195706601283614000430, 3.39229902308404054209432818102, 3.90868214594542927641564063072, 5.04911992710713684753711530551, 5.83488728820976416365292429234, 7.48422665484770251711252216365, 8.150201953600577860913937933968, 8.727985839479846086002192914075, 10.30160929331047313107524908295, 11.16891789193882987488113909891

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