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Properties

Label 2-864-12.11-c3-0-6
Degree $2$
Conductor $864$
Sign $-0.707 - 0.707i$
Analytic cond. $50.9776$
Root an. cond. $7.13986$
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  + 11.4i·5-s − 5.04i·7-s + 13.8·11-s − 17.8·13-s + 32.0i·17-s − 6.50i·19-s + 100.·23-s − 5.28·25-s + 136. i·29-s + 136. i·31-s + 57.6·35-s − 319.·37-s − 38.8i·41-s + 78.3i·43-s + 64.1·47-s + ⋯
L(s)  = 1  + 1.02i·5-s − 0.272i·7-s + 0.379·11-s − 0.380·13-s + 0.457i·17-s − 0.0785i·19-s + 0.906·23-s − 0.0422·25-s + 0.873i·29-s + 0.791i·31-s + 0.278·35-s − 1.41·37-s − 0.147i·41-s + 0.277i·43-s + 0.199·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(50.9776\)
Root analytic conductor: \(7.13986\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :3/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.287049721\)
\(L(\frac12)\) \(\approx\) \(1.287049721\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 11.4iT - 125T^{2} \)
7 \( 1 + 5.04iT - 343T^{2} \)
11 \( 1 - 13.8T + 1.33e3T^{2} \)
13 \( 1 + 17.8T + 2.19e3T^{2} \)
17 \( 1 - 32.0iT - 4.91e3T^{2} \)
19 \( 1 + 6.50iT - 6.85e3T^{2} \)
23 \( 1 - 100.T + 1.21e4T^{2} \)
29 \( 1 - 136. iT - 2.43e4T^{2} \)
31 \( 1 - 136. iT - 2.97e4T^{2} \)
37 \( 1 + 319.T + 5.06e4T^{2} \)
41 \( 1 + 38.8iT - 6.89e4T^{2} \)
43 \( 1 - 78.3iT - 7.95e4T^{2} \)
47 \( 1 - 64.1T + 1.03e5T^{2} \)
53 \( 1 + 91.5iT - 1.48e5T^{2} \)
59 \( 1 + 123.T + 2.05e5T^{2} \)
61 \( 1 + 563.T + 2.26e5T^{2} \)
67 \( 1 - 874. iT - 3.00e5T^{2} \)
71 \( 1 + 581.T + 3.57e5T^{2} \)
73 \( 1 + 509.T + 3.89e5T^{2} \)
79 \( 1 - 140. iT - 4.93e5T^{2} \)
83 \( 1 + 810.T + 5.71e5T^{2} \)
89 \( 1 + 963. iT - 7.04e5T^{2} \)
97 \( 1 + 1.23e3T + 9.12e5T^{2} \)
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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

−10.40086251675036087994669669599, −9.230376955229191212026805552580, −8.476358634147078691038472341695, −7.17737412433120391024552010699, −6.94695818277665855736299003794, −5.81912272799031460861798238852, −4.73839037932025667104371083377, −3.57837331278877562248634558777, −2.74742768118484776986504352824, −1.36951789679988686896188422706, 0.34346668844074272273911240369, 1.55401038249165476609252173008, 2.85836366305820904854874992549, 4.17596409416712346859831113278, 4.99413745100488576564364684995, 5.81969692741278806051629221041, 6.92901928650661695815715869905, 7.82818463015825180960245866981, 8.819706619599441157131847276409, 9.247342457227924223875432978934

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