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Properties

Label 2-567-63.32-c0-0-0
Degree $2$
Conductor $567$
Sign $0.110 + 0.993i$
Analytic cond. $0.282969$
Root an. cond. $0.531949$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + (−1 − 1.73i)19-s + 25-s + (−0.499 + 0.866i)28-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)43-s + (−0.499 + 0.866i)49-s − 0.999·52-s + (0.5 − 0.866i)61-s + 0.999·64-s + (0.5 + 0.866i)67-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + (−1 − 1.73i)19-s + 25-s + (−0.499 + 0.866i)28-s + (0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 + 0.866i)43-s + (−0.499 + 0.866i)49-s − 0.999·52-s + (0.5 − 0.866i)61-s + 0.999·64-s + (0.5 + 0.866i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $0.110 + 0.993i$
Analytic conductor: \(0.282969\)
Root analytic conductor: \(0.531949\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :0),\ 0.110 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7385663214\)
\(L(\frac12)\) \(\approx\) \(0.7385663214\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.66441396906744075063280721368, −10.00243066083635070250225121219, −9.068459377304977101431518781175, −8.270703238010723335260476607775, −6.93619170063442069772384538846, −6.29299669279842572482419776093, −5.07886276951766432565279938684, −4.26071148416567980616855146364, −2.89430962014054904977081826944, −0.950768804553824720539928995946, 2.25104915072872392365362427961, 3.53943531403397835777709993665, 4.39770798204362694050983804405, 5.73798975340171849314763020053, 6.60627409038309433457391329603, 7.77845934134753511236041373744, 8.638758592123385911157950008101, 9.182887856289517560094433755798, 10.18018999179458385837757307519, 11.30874270988661262905825516018

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