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Properties

Label 2-567-63.41-c1-0-24
Degree $2$
Conductor $567$
Sign $-0.873 + 0.486i$
Analytic cond. $4.52751$
Root an. cond. $2.12779$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)4-s + (−2.5 + 0.866i)7-s + (−4.5 − 2.59i)13-s + (−1.99 − 3.46i)16-s − 5.19i·19-s + (2.5 + 4.33i)25-s + (1.00 − 5.19i)28-s + (−9 − 5.19i)31-s − 11·37-s + (4 + 6.92i)43-s + (5.5 − 4.33i)49-s + (9 − 5.19i)52-s + (−13.5 + 7.79i)61-s + 7.99·64-s + (−2.5 + 4.33i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (−0.944 + 0.327i)7-s + (−1.24 − 0.720i)13-s + (−0.499 − 0.866i)16-s − 1.19i·19-s + (0.5 + 0.866i)25-s + (0.188 − 0.981i)28-s + (−1.61 − 0.933i)31-s − 1.80·37-s + (0.609 + 1.05i)43-s + (0.785 − 0.618i)49-s + (1.24 − 0.720i)52-s + (−1.72 + 0.997i)61-s + 0.999·64-s + (−0.305 + 0.529i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.873 + 0.486i$
Analytic conductor: \(4.52751\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 567,\ (\ :1/2),\ -0.873 + 0.486i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.5 + 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (9 + 5.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (13.5 - 7.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.24074203240994543128786766143, −9.305376816106342791127316224491, −8.842825534910536233909152303612, −7.53945045427943666969500758443, −7.05461364501904972183295120734, −5.64723959938980570543294405003, −4.70008175534971652701706177576, −3.43273011617679706088278541185, −2.61219848414164231881853277085, 0, 1.86922413472460957671198090690, 3.50247578312904356705651393375, 4.61128609651364563231209521890, 5.58704733587315775942695129740, 6.57565482360371584470628147888, 7.37099920628446422811169936807, 8.757651942940621712886122483012, 9.442854057083816748327188685896, 10.22688009655754631698093061475

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