Properties

Label 2-567-21.20-c3-0-80
Degree $2$
Conductor $567$
Sign $-0.612 - 0.790i$
Analytic cond. $33.4540$
Root an. cond. $5.78395$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.90i·2-s − 7.26·4-s + 11.6·5-s + (−14.6 + 11.3i)7-s − 2.86i·8-s − 45.5i·10-s − 47.6i·11-s + 69.5i·13-s + (44.3 + 57.2i)14-s − 69.3·16-s − 25.6·17-s − 41.1i·19-s − 84.6·20-s − 186.·22-s − 168. i·23-s + ⋯
L(s)  = 1  − 1.38i·2-s − 0.908·4-s + 1.04·5-s + (−0.790 + 0.612i)7-s − 0.126i·8-s − 1.43i·10-s − 1.30i·11-s + 1.48i·13-s + (0.845 + 1.09i)14-s − 1.08·16-s − 0.366·17-s − 0.496i·19-s − 0.946·20-s − 1.80·22-s − 1.52i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(567\)    =    \(3^{4} \cdot 7\)
Sign: $-0.612 - 0.790i$
Analytic conductor: \(33.4540\)
Root analytic conductor: \(5.78395\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{567} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 567,\ (\ :3/2),\ -0.612 - 0.790i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9437252304\)
\(L(\frac12)\) \(\approx\) \(0.9437252304\)
\(L(\frac{5}{2})\) 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 \( 1 + (14.6 - 11.3i)T \)
good2 \( 1 + 3.90iT - 8T^{2} \)
5 \( 1 - 11.6T + 125T^{2} \)
11 \( 1 + 47.6iT - 1.33e3T^{2} \)
13 \( 1 - 69.5iT - 2.19e3T^{2} \)
17 \( 1 + 25.6T + 4.91e3T^{2} \)
19 \( 1 + 41.1iT - 6.85e3T^{2} \)
23 \( 1 + 168. iT - 1.21e4T^{2} \)
29 \( 1 - 32.5iT - 2.43e4T^{2} \)
31 \( 1 + 171. iT - 2.97e4T^{2} \)
37 \( 1 + 3.86T + 5.06e4T^{2} \)
41 \( 1 + 337.T + 6.89e4T^{2} \)
43 \( 1 + 414.T + 7.95e4T^{2} \)
47 \( 1 + 148.T + 1.03e5T^{2} \)
53 \( 1 - 59.7iT - 1.48e5T^{2} \)
59 \( 1 + 564.T + 2.05e5T^{2} \)
61 \( 1 + 660. iT - 2.26e5T^{2} \)
67 \( 1 + 291.T + 3.00e5T^{2} \)
71 \( 1 - 3.05iT - 3.57e5T^{2} \)
73 \( 1 - 506. iT - 3.89e5T^{2} \)
79 \( 1 - 193.T + 4.93e5T^{2} \)
83 \( 1 - 432.T + 5.71e5T^{2} \)
89 \( 1 - 924.T + 7.04e5T^{2} \)
97 \( 1 - 1.08e3iT - 9.12e5T^{2} \)
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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

−9.833019175813533091393442342465, −9.205976064644593668433800548250, −8.574907196230402265425197911722, −6.61873349974063528168675476946, −6.25100817873292693198048698518, −4.85691124555219094197842728519, −3.59127683772995808213883661026, −2.59634767282157426883871898720, −1.78870457867573070578061658598, −0.25408666733573812516726488198, 1.75947109092752303214878901759, 3.28328134717065877238103518804, 4.81780502068922322693060898767, 5.62221754523388537373389424279, 6.42472305753284341052324053628, 7.22522638268877380339779061414, 7.926957685501951153257773106775, 9.103650366073357038169030834295, 9.958155326298943142537647336637, 10.42325959009957843642506271091

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