Properties

Label 2-864-4.3-c2-0-3
Degree $2$
Conductor $864$
Sign $-0.707 - 0.707i$
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  + 3.22·5-s − 6.57i·7-s + 13.8i·11-s − 19.0·13-s − 23.0·17-s + 18.8i·19-s − 13.5i·23-s − 14.5·25-s − 0.752·29-s + 46.4i·31-s − 21.1i·35-s − 2.68·37-s + 34.1·41-s + 20.9i·43-s + 15.4i·47-s + ⋯
L(s)  = 1  + 0.645·5-s − 0.938i·7-s + 1.26i·11-s − 1.46·13-s − 1.35·17-s + 0.991i·19-s − 0.591i·23-s − 0.583·25-s − 0.0259·29-s + 1.49i·31-s − 0.605i·35-s − 0.0724·37-s + 0.832·41-s + 0.486i·43-s + 0.327i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(23.5422\)
Root analytic conductor: \(4.85204\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6710884168\)
\(L(\frac12)\) \(\approx\) \(0.6710884168\)
\(L(2)\) 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 \( 1 - 3.22T + 25T^{2} \)
7 \( 1 + 6.57iT - 49T^{2} \)
11 \( 1 - 13.8iT - 121T^{2} \)
13 \( 1 + 19.0T + 169T^{2} \)
17 \( 1 + 23.0T + 289T^{2} \)
19 \( 1 - 18.8iT - 361T^{2} \)
23 \( 1 + 13.5iT - 529T^{2} \)
29 \( 1 + 0.752T + 841T^{2} \)
31 \( 1 - 46.4iT - 961T^{2} \)
37 \( 1 + 2.68T + 1.36e3T^{2} \)
41 \( 1 - 34.1T + 1.68e3T^{2} \)
43 \( 1 - 20.9iT - 1.84e3T^{2} \)
47 \( 1 - 15.4iT - 2.20e3T^{2} \)
53 \( 1 - 46.8T + 2.80e3T^{2} \)
59 \( 1 - 40.4iT - 3.48e3T^{2} \)
61 \( 1 + 105.T + 3.72e3T^{2} \)
67 \( 1 - 27.9iT - 4.48e3T^{2} \)
71 \( 1 + 24.0iT - 5.04e3T^{2} \)
73 \( 1 + 120.T + 5.32e3T^{2} \)
79 \( 1 - 95.6iT - 6.24e3T^{2} \)
83 \( 1 + 115. iT - 6.88e3T^{2} \)
89 \( 1 + 169.T + 7.92e3T^{2} \)
97 \( 1 + 93.1T + 9.40e3T^{2} \)
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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

−10.14290457645645746636580332215, −9.689580303527192600181648016443, −8.689574652394480420881146395997, −7.44405753663031079766884768756, −7.05998846382149993196432025560, −5.99011348718323664612090291601, −4.77452046208231275807539345199, −4.21241558659802511578647993113, −2.62158357252990220999774832749, −1.63357783994312492565393619971, 0.20069266792465771188868627119, 2.12429536201764548637511788926, 2.80695185863751909509632940979, 4.31792525216372395775667378277, 5.40939147421134349912422229756, 6.00028001347191085524179937952, 7.01120055445478808193079714939, 8.032379553295088826391016300275, 9.101685831847850062816567445181, 9.357554391247993592443157159496

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