Properties

Degree $2$
Conductor $567$
Sign $0.939 - 0.342i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.500 + 0.866i)4-s + (−1 − 1.73i)5-s + (0.5 − 0.866i)7-s − 3·8-s + 1.99·10-s + (2 − 3.46i)11-s + (1 + 1.73i)13-s + (0.499 + 0.866i)14-s + (0.500 − 0.866i)16-s + 6·17-s + 4·19-s + (0.999 − 1.73i)20-s + (1.99 + 3.46i)22-s + (0.500 − 0.866i)25-s − 1.99·26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s + (−0.447 − 0.774i)5-s + (0.188 − 0.327i)7-s − 1.06·8-s + 0.632·10-s + (0.603 − 1.04i)11-s + (0.277 + 0.480i)13-s + (0.133 + 0.231i)14-s + (0.125 − 0.216i)16-s + 1.45·17-s + 0.917·19-s + (0.223 − 0.387i)20-s + (0.426 + 0.738i)22-s + (0.100 − 0.173i)25-s − 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.939 - 0.342i$
Motivic weight: \(1\)
Character: $\chi_{567} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25463 + 0.221226i\)
\(L(\frac12)\) \(\approx\) \(1.25463 + 0.221226i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + (9 - 15.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.92450201753723108839159723850, −9.591201153340732674910822238006, −8.828559430282547778411203523050, −8.083027083479950121941719978622, −7.43596156329780616969054871519, −6.34751108222972485962694119457, −5.43133796436705680912532916802, −4.05831821280693449248314312925, −3.13902494119266651398648008825, −1.01961052068270479706967825022, 1.28787422604450895769164124918, 2.67897545623779410770913438795, 3.64760841523237647680286086877, 5.19548638417327909492650328365, 6.16140226457456699023422821152, 7.18563224637505461237775186972, 7.986067018453158391574882202089, 9.338145666460537515354315811243, 9.858738689022464400491726851907, 10.70731250707372895790968129842

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