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Properties

Label 2-864-24.5-c2-0-27
Degree $2$
Conductor $864$
Sign $0.626 + 0.779i$
Analytic cond. $23.5422$
Root an. cond. $4.85204$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.79·5-s + 10.8·7-s + 0.420·11-s − 21.1i·13-s − 30.5i·17-s − 13.1i·19-s + 22.5i·23-s − 10.5·25-s − 15.4·29-s + 22.7·31-s + 41.0·35-s + 15.0i·37-s + 10.4i·41-s − 6.13i·43-s + 36.2i·47-s + ⋯
L(s)  = 1  + 0.759·5-s + 1.54·7-s + 0.0382·11-s − 1.62i·13-s − 1.79i·17-s − 0.691i·19-s + 0.980i·23-s − 0.423·25-s − 0.531·29-s + 0.732·31-s + 1.17·35-s + 0.406i·37-s + 0.254i·41-s − 0.142i·43-s + 0.770i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.626 + 0.779i$
Analytic conductor: \(23.5422\)
Root analytic conductor: \(4.85204\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1),\ 0.626 + 0.779i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.514641136\)
\(L(\frac12)\) \(\approx\) \(2.514641136\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 3.79T + 25T^{2} \)
7 \( 1 - 10.8T + 49T^{2} \)
11 \( 1 - 0.420T + 121T^{2} \)
13 \( 1 + 21.1iT - 169T^{2} \)
17 \( 1 + 30.5iT - 289T^{2} \)
19 \( 1 + 13.1iT - 361T^{2} \)
23 \( 1 - 22.5iT - 529T^{2} \)
29 \( 1 + 15.4T + 841T^{2} \)
31 \( 1 - 22.7T + 961T^{2} \)
37 \( 1 - 15.0iT - 1.36e3T^{2} \)
41 \( 1 - 10.4iT - 1.68e3T^{2} \)
43 \( 1 + 6.13iT - 1.84e3T^{2} \)
47 \( 1 - 36.2iT - 2.20e3T^{2} \)
53 \( 1 + 22.1T + 2.80e3T^{2} \)
59 \( 1 + 101.T + 3.48e3T^{2} \)
61 \( 1 + 94.9iT - 3.72e3T^{2} \)
67 \( 1 + 90.8iT - 4.48e3T^{2} \)
71 \( 1 - 63.5iT - 5.04e3T^{2} \)
73 \( 1 - 81.1T + 5.32e3T^{2} \)
79 \( 1 - 38.0T + 6.24e3T^{2} \)
83 \( 1 - 111.T + 6.88e3T^{2} \)
89 \( 1 - 70.0iT - 7.92e3T^{2} \)
97 \( 1 - 144.T + 9.40e3T^{2} \)
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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

−9.766399628042383422785719563984, −9.133256003149554920184396397191, −7.915694528222489831597149413118, −7.60910483796089610588658068136, −6.26145096460754084435129295605, −5.18225539372939554376529407600, −4.88516677014278220127961736965, −3.21576370907211963372992069231, −2.12306927751757402615278975838, −0.868509716819243490267274627232, 1.58337647117500837594788947961, 2.06970862241595621924164299179, 3.95344568614688764481435283420, 4.66193978374645556770654987963, 5.79014043497297665718818064337, 6.48579499152632203724551965076, 7.67196702709084648374566472949, 8.446205511734035485560729052879, 9.135739486645023950385117525846, 10.20820182050050511918295121010

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