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Properties

Label 2-567-63.20-c1-0-24
Degree $2$
Conductor $567$
Sign $-0.0155 - 0.999i$
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  + (−1.93 − 1.11i)2-s + (1.5 + 2.59i)4-s + (−1.93 − 3.35i)5-s + (−0.5 − 2.59i)7-s − 2.23i·8-s + 8.66i·10-s + (−1.93 − 1.11i)11-s + (−3 + 1.73i)13-s + (−1.93 + 5.59i)14-s + (0.499 − 0.866i)16-s − 5.19i·19-s + (5.80 − 10.0i)20-s + (2.5 + 4.33i)22-s + (1.93 − 1.11i)23-s + (−5.00 + 8.66i)25-s + 7.74·26-s + ⋯
L(s)  = 1  + (−1.36 − 0.790i)2-s + (0.750 + 1.29i)4-s + (−0.866 − 1.50i)5-s + (−0.188 − 0.981i)7-s − 0.790i·8-s + 2.73i·10-s + (−0.583 − 0.337i)11-s + (−0.832 + 0.480i)13-s + (−0.517 + 1.49i)14-s + (0.124 − 0.216i)16-s − 1.19i·19-s + (1.29 − 2.25i)20-s + (0.533 + 0.923i)22-s + (0.403 − 0.233i)23-s + (−1.00 + 1.73i)25-s + 1.51·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0155 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0155 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.0155 - 0.999i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.0155 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117482 + 0.119329i\)
\(L(\frac12)\) \(\approx\) \(0.117482 + 0.119329i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 + 1.11i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (1.93 + 3.35i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.93 + 1.11i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (-1.93 + 1.11i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.87 - 2.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (1.93 + 3.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.87 - 6.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.94iT - 53T^{2} \)
59 \( 1 + (-3.87 - 6.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.87 - 6.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (12 + 6.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.00594671116521917235045913853, −9.172798094865438127365393850346, −8.543247378461032768292799207288, −7.75140953095226141983271967862, −7.02032549961440959918061110795, −5.09696565639077180913743624516, −4.28254387845320465658403683424, −2.85320460877019664862127618499, −1.17217951323115637267547343748, −0.16878747350830602566947762291, 2.34803643237193973464140030634, 3.50076372807318836771432365393, 5.31723219767917975453818055248, 6.44996414154175575816648554645, 7.07532262974005055142985997225, 7.968513796623174273121482163540, 8.404545989003597696938160414465, 9.909202867639145241723622490118, 10.04445509222088144373060044273, 11.12923322248808784252893868768

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