Properties

Label 2-567-63.4-c1-0-19
Degree $2$
Conductor $567$
Sign $0.296 + 0.954i$
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 − 1.73i)2-s + (−0.999 − 1.73i)4-s + 2·5-s + (0.5 + 2.59i)7-s + (2 − 3.46i)10-s + 2·11-s + (−0.5 + 0.866i)13-s + (5 + 1.73i)14-s + (1.99 − 3.46i)16-s + (−0.5 − 0.866i)19-s + (−1.99 − 3.46i)20-s + (2 − 3.46i)22-s − 25-s + (0.999 + 1.73i)26-s + (4 − 3.46i)28-s + (2 + 3.46i)29-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + 0.894·5-s + (0.188 + 0.981i)7-s + (0.632 − 1.09i)10-s + 0.603·11-s + (−0.138 + 0.240i)13-s + (1.33 + 0.462i)14-s + (0.499 − 0.866i)16-s + (−0.114 − 0.198i)19-s + (−0.447 − 0.774i)20-s + (0.426 − 0.738i)22-s − 0.200·25-s + (0.196 + 0.339i)26-s + (0.755 − 0.654i)28-s + (0.371 + 0.643i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.296 + 0.954i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.296 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06141 - 1.51807i\)
\(L(\frac12)\) \(\approx\) \(2.06141 - 1.51807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.79798003878440369625481114458, −9.743085392573748199101155708867, −9.269317929588366839494968781945, −8.081369183952875929268387637442, −6.67401132926729745561805558542, −5.65402413059705802940052578498, −4.84646998447341354601751317251, −3.64715336700452918307007466874, −2.45581462611777429379153939795, −1.66496494276734446870788956236, 1.62440466705983268975251950748, 3.56521460178624938428373487826, 4.60396764432397262044816581145, 5.49715026122201957753385216282, 6.39695686068021408051194423395, 7.07624039652029814015629422302, 7.950808660336606371306068858911, 8.995207274493933631245452393988, 10.13300321262547340111441293056, 10.68895164438280465730206340110

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