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Properties

Label 2-864-24.5-c2-0-11
Degree $2$
Conductor $864$
Sign $0.299 - 0.954i$
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  + 2.62·5-s − 0.432·7-s − 5.01·11-s − 1.56i·13-s + 19.8i·17-s + 24.1i·19-s + 3.07i·23-s − 18.1·25-s + 34.4·29-s + 43.4·31-s − 1.13·35-s − 52.9i·37-s + 56.7i·41-s + 27.6i·43-s + 83.7i·47-s + ⋯
L(s)  = 1  + 0.524·5-s − 0.0617·7-s − 0.455·11-s − 0.120i·13-s + 1.16i·17-s + 1.27i·19-s + 0.133i·23-s − 0.725·25-s + 1.18·29-s + 1.40·31-s − 0.0323·35-s − 1.43i·37-s + 1.38i·41-s + 0.643i·43-s + 1.78i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.299 - 0.954i$
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: \((2,\ 864,\ (\ :1),\ 0.299 - 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.686274048\)
\(L(\frac12)\) \(\approx\) \(1.686274048\)
\(L(2)\) 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 \( 1 - 2.62T + 25T^{2} \)
7 \( 1 + 0.432T + 49T^{2} \)
11 \( 1 + 5.01T + 121T^{2} \)
13 \( 1 + 1.56iT - 169T^{2} \)
17 \( 1 - 19.8iT - 289T^{2} \)
19 \( 1 - 24.1iT - 361T^{2} \)
23 \( 1 - 3.07iT - 529T^{2} \)
29 \( 1 - 34.4T + 841T^{2} \)
31 \( 1 - 43.4T + 961T^{2} \)
37 \( 1 + 52.9iT - 1.36e3T^{2} \)
41 \( 1 - 56.7iT - 1.68e3T^{2} \)
43 \( 1 - 27.6iT - 1.84e3T^{2} \)
47 \( 1 - 83.7iT - 2.20e3T^{2} \)
53 \( 1 + 41.4T + 2.80e3T^{2} \)
59 \( 1 - 74.2T + 3.48e3T^{2} \)
61 \( 1 + 28.7iT - 3.72e3T^{2} \)
67 \( 1 + 33.8iT - 4.48e3T^{2} \)
71 \( 1 - 104. iT - 5.04e3T^{2} \)
73 \( 1 - 53.2T + 5.32e3T^{2} \)
79 \( 1 + 51.8T + 6.24e3T^{2} \)
83 \( 1 - 76.3T + 6.88e3T^{2} \)
89 \( 1 - 131. iT - 7.92e3T^{2} \)
97 \( 1 - 68.9T + 9.40e3T^{2} \)
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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

−10.08349121680170775419049142440, −9.497242474235615170811808718150, −8.228512358149860525005570665266, −7.87162118396367299228354379902, −6.41696868821383846386367680007, −5.95676100611969858257878631306, −4.83132927716382806850750872421, −3.75413754506119696595873453579, −2.54920519794625619941117306657, −1.33313699081606985017159419705, 0.57899067726561530975036566742, 2.20745359833307905267967874391, 3.12740801928643211673186911386, 4.59968418492404274140838575072, 5.27475464607908464747510882304, 6.42985626986713154357438635465, 7.10036011467307611303786128589, 8.192349302640229956363736036624, 8.990110603607584351245718912067, 9.883484290290993737132043175910

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